Transcript for:
Understanding Taylor and McLaurin Series

all right putting everything together now we're going to talk about Taylor series and mcloren series now what this is what this basic idea is all about is how to represent a function with a power series let's go through some of the theory behind this all right so firstly if a function f has a power series or let's let's do it this way suppose suppose the function f has a power series repres well then a few things are true the first thing we we know and let's suppose this is at the point C the first thing we know is true is that F ofx would equal a series from n = 0 to Infinity of a subn x - c to the N you see where the x - c is coming from if this is a power series representation at C well then we can say okay it's going to be x - c to some Power Times some sequence of terms and then we add them all up together you guys okay with that so far that is a power Series right in fact that's a general form of a power series even if C is zero that's still the power series that represent that so if we have a function that can be represented by a power series it's going to look like like this somehow show FP right with that one that's just all we're saying it says if F has a power representation it's going to look like that now what we're going to state next is this if this is true if we have a power series representation because we have these terms the the the sequence of terms where we get our series from all of these as subn are going to be constants true or false which means the x is coefficients yeah as n goes from Z to Infinity then we're going to have X basically x to the 0 x to the 1 x to the 2 x to the 3 x to the N correct what that means is that for sure if this means makes a power series we're going to have a whole bunch of terms right wherever we stop is going to be the nth term does that make sense to you that's going to be x to the N what that means is that the F to the N derivative must exist it's got to be there because what we're going to end up getting is whole bunch of coefficients these are constants fractions I don't care what they R it's going to be something time x to the blank power and that n power means that you could take a whole bunch of derivatives couldn't you hello yes no if I had like three terms I could take up to a third derivative and it's going to be there it's going to be a constant actually well that's going to be you say okay bring down the power subtract one from it that's first derivative bring down the power subtract one from it that's a second dtive bring down the third power subtract one from it that's a fourth third derivative and so on you can do that up to the nth derivative because there's going to be a power n somewhere in your power Series in your list of terms does that make sense so then we say all right well then the nth derivative of f exists at C and the way we write that is f to the well not F to the I'm sorry um the N derivative of F at C that's what that says I'll go through it one more time just so that that makes sense to you and then I'm going to show you what we can do with this thing this is going to be interesting this is where we're going to get our tailor series from so here's the idea one one more time if F has a power series representation at C then this is for sure true then F can be represented by a power series not a problem a power series just X to some power if we have X to some power n and we list out all of these terms and we start taking derivatives then the first derivative is going to exist and the second one and the third one and the fourth one all the way up into the nth derivative because what's going to happen is that our first term is going to go to zero it's constant our second term becomes a constant therefore when I take my next derivative that term goes to zero that's going to happen until we get down to our nth derivative our nth term ends end up turning into a constant at our n derivative does that make sense to you all the rest of them are going to be something like x - c to the whatever the the next term after the N1 n plus one that's going to be xus C to the 1 and then to the second then to the third then the fourth but we can take up to the nth derivative and it's going to exist somewhere at C just means hey plug in C not a problem show fans you feel okay that this is going to exist and this is going to make a little bit more sense when we go through the next thing I'm talking about the next thing I want to find out and I'm going to write St up here I'm going to erase it in just a minute this is just for to show you what what this is the last thing is I want to find out what a subn is in terms of my derivative in terms of that and we're going to do it this way you guys ready for it so we're going to come up with what a subn is the idea is what I'm trying to do is figure out basically a formula that's based on my function that I can put right here because if I can do that if I can base this on my function well then I can start building a power series from a specific function so here's the idea on this you know what I paper did it separately let's start with f ofx thank you you if F ofx has a power series representation then it's going to be represented by this thing in general it's going to be represented by this give me the first term of this power series a sub Z and it would be this to the zero give you one be a sub Z then I would get what's the next ter to the plus a uhhuh x - c 2 plus a plus blah blah blah blah blah all the way down until we get to n a sub a subn xus c to the good you guys okay with that one now this says we stop at n but in reality this thing would keep on going forever and ever and ever and ever and ever ever never ever because that thing does not end it goes to Infinity are you with me so far so here's the point I was trying to make earlier maybe you can see a little bit better now do you understand that if this is my power series representation it basically acts like a polinomial here it's going to be X is some power these are all constants that that they're coefficients of whatever X term we have what that means is could I take a first derivative sure could I take a second derivative could I take up to an nth derivative abolutely all it's going to happen is I take a d all these things n times or whatever n is five times seven times a million times times who cares if I have a millionth term then I can take a millionth derivative and then I can plug C in does that make sense now this going to get kind of interesting for us let's start taking derivatives if I take the first derivative I'm going to erase that if I take the first derivative frime of x help me out with this what do I get remember we did this in the last section this is why we we just did that is so that you understand that you can take a derivative of a power series that's why we did it CU now we're going to use the last idea that we had in the previous section right now so take your derivative what happens to a sub z0 that's gone what happens to a sub1 x - c what's that become a good plus what 2 a 2 oh good what Happ why why the two what are we going to get there so 2 * a sub 2 x - c to the first Power very good plus 3 a sub 3 a 3 x - c to the power now this is the one that I really want to focus on this one you see here's why I want to focus on that the what we get got from the nth term why I want to focus on that one is because when we get down to nth derivative do you understand all these things are ultimately going to go to zero they're all going to become constants so I'm really focused on this guy so what happens here oops and then we would continue you all okay with that one that's the one I want you to focus on to see what happens with this okay let's continue we're going to do like uh four derivatives three or four derivatives so just to get a really strong pattern here so let's do fpre of X okay tell me what happens come on you guys got it what happens here what happens here really what happens is you bring down the one don't you M you really get 2 * 1 do you see it a sorry a sub 2 x - C 0 but that's going to go away so I get 2 * 1 * a sub 2 2 A 2 yep now I don't want you to simplify because we're looking for a pattern here does that make sense for pattern okay next up it's three don't give me six I don't want six with this stuff we're really looking for a pattern here so it's 3 * 2 2 * a sub 3 x - c to the plus okay now remember this is the one I want to focus on right here what's going to happen good you'll bring down the N minus one then you multip byus two very good we okay with that so far okay it's going to keep on going let's do this is the last one that we'll do because the pattern is going to be apparent after this if we did a third derivative what happens to this term where's it go that's all that's all a constant that's gone does that make sense this one would be 3 * 2 * * 1 a sub 3 x - c to the Z that's going to go away that's going to be one are you with me you sure you're with me okay plus blah blah blah do you notice that what happens is that we're going to end up getting a constant for every single term that we have you with me so check this out when we did a first derivative what I ended with was a sub one term as a constant are you listening when I did my second derivative I ended with my a sub 2 term as a constant do you see it a sub3 had an X with it when I did my third derivative I end with my a sub3 term as a constant my a sub4 term would have an X to some power does that make sense okay now let's do our our next derivative uh what am I going to get from this term as a derivative a sub very good x c to plus blah blah blah blah blah forever and ever ever now we're going to make a big jump we're going to go down here and do the nth derivative firstly does this make sense to you where all these things are coming from hello yes no are you sure we're just doing derivatives we're doing chain rule no problem deriv inside's always one so we got it easy here what I'm trying to get you to see is the pattern that's taking place what we saw is that okay well when we work this down we got 21 we got 321 we're going to have 4 3 2 1 we're going to have 5 4 3 2 1 we're going to have 6 54 3 21 we're going to have n factorial somewhere in this problem do you see it also it says right here we have n we have n * n -1 n * n -1 * N - 2 that's a factorial idea start with 7 multiply by 6 multiply by 5 multi start with n multiply by nus 1 N - 2 and all the way down until you get to one are you with me also what's happening I wanted you to see this first derivative says a sub 1 is your constant term second dtive a sub2 is your constant term third derivative a sub3 is your constant term nth derivative a subn is your constant term so ultimately what we're going to get down to is every term before it is going to go to zero the a sub one term goes to zero a sub 2 term goes to zero a sub 3 term goes to zero a sub n minus one term is going to go to zero if we take the nth derivative do you get that at the nth derivative what's going to happen here is you've subtracted n minus n you you'll have n minus n there that would give you the zero does that make sense so we'd bring it down we'd have an N then we'd have an N minus one then we'd have an N minus 2 then we'd have multiply all the way down until you get 3 2 1 and then youd have an H sub B does that make sense are you sure it make sense we would have x - c to the n- n and then we'd keep on going this would be a blah blah blah blah blah but I want to talk about this for a bit I'll try to I'll try to have this be really stuck in your head because this is what this is what we're going to get our Taylor series from just a moment okay so so bear with me for just a second if you were to take the nth derivative of a function like this one if you were to do that and nth derivative means not the first not the second not the third not the fourth not the fifth not the sixth but do it forever and then go forever but do it to a certain point n what's going to happen is that every Power every term with a power less than n it's a polinomial that's going to go to zero do you get that every term is going to go to zero except for the nth term at the nth term you will have done so many derivatives the first thing you do is the n then you multiply by n minus one then nus 2 then nus 3 all the way down until you get to one does that make sense now the N minus n that's going to give you zero what happens to this final little piece here where's that where's that go this goes to one anything to the zero power is one so this is going to be gone so what we know is that if I take an N derivative what's going to happen here is because I'm taking this down every time and saying hey one more derivative I bring down the N minus 2 hey look at that one more derivative look at this my next derivative I'd bring down nus 3 do you get it my next der I bring down nus 4 nus 5 all the way until I get to N - n -1 and that would give me my 2 and then my 1 and then I'd have a subn but then I have x - c to the 0 because at my nth derivative it'd be n minus n and my n minus n would give me zero anything to Zer power gives me one this would be my very first term so hands feel okay with that now this doesn't stop but when I keep going every term past this is going to have some bunch of crap x - c to some power that's very un mathematical but I'm trying to get you the idea here does that make sense to you are you sure so my very my derivative my very first term is going to be n * all this junk and then a subn with no x- C but every term after that is going to have an xus C do you understand why this does not have an xus C do you understand what all these ones do anything past the the nth term is going to have an x -1 to one more power so when I take an N derivative I still have a power there that's not zero does does that make sense now let's write this better what is all that very good now what I care about what I don't care about I don't care about F the the nth derivative of F at The X what I want to do is this at C so we're going to make one little jump if this is true if the N root of f ofx is all this junk then if I want to find the N root of F at C let's think about what's going to happen please please bear in mind this is why I just asked the question do you understand that there's no x- n here or sorry uh xus what is that c c there's no x- C here but every other term is going to have an xus C do you get that yes do you understand that F of c means I plug in C for X yeah so if every one of these terms has an x minus C with it and I plug in C for my X what's cus C everything else is going to go to zero and the only thing I'm left with is this guy n factorial time B of N and that's it did that make sense to you it's kind of weird right but this is the idea now here's what's what's awesome about this can you solve for a subn mhm very easily this is where we're going to make the jump so I hope you caught on to this this if every other term has x- C when I plug in C C- C is Z all these things are gone this does not have any X's at all it's just a constant this is the first constant when I plug in C everything else is gone except for that constant this is n factorial this is still a subn let's solve for a subn if I solve for a subn I get a subn equals let's just divide both sides F the nth root of F at C / n factorial that ladies and gentlemen was what I was looking for over here well what that means is that if I have just solv for the as subn term this is really kind of a neat idea do you understand that this was all in general right I made no specific case about this whatsoever I said if this has a power representation we have this we can take derivatives of it no problem of course we can let's do derivatives this right here would be the nth der of F at X if I plug in C which wherever my power is centered I can do that if I plug in C everything's going to disappear except for this very last term this very last term it has a subn in it so if I want to solve for a of n i just divide what that means is that what we can do is change this power series becomes this power series why don't you tell me instead of having a subn here what am I going to put what am I going to put the N derivative of F at C it's hard to it looks weird it's Weir to say but the N derivative of F at C the nth derivative of F at C uh-huh and then what Over N factorial very good and then what okay I want to make sure that if there's any questions I answer them right now do you all do yall see what I'm talking about do you see that what this thing becomes what this is the same okay what this thing becomes is if I know this is true which I just proved it that it is if I know this is true instead of a subn I can have the N derivative of F at C divided n factorial and that is what goes right there should fans if that one made a little bit of sense to you okay what this is is a tailor Series this right here is the tailor series any questions question on this at all because I'm going to race this whole side you sure did it make sense to you just some basic derivatives maybe not so basic so this thing this series from 0 to Infinity of the nth derivative of F at C Over N factorial - n- * x - c to n let's look at how this would look okay just like the first few terms what's the first term let's plug what would you plug in to find the first term plug in zero okay so um if you plugged in zero what is it's not F to the 0o it's the you don't take a derivative of f yet you don't have it you take zero derivatives that's what that little N means it means if that was a one you take the first derivative if that was a two you take the second der that's a zero you don't take the derivative does that make sense x - C2 though what would that become that would be one all over Z factorial 0 factorial if you don't remember is defined as one so this would simply be F of C you don't take the derivative of f you plug in C that's the very first term here does that make sense to you the next term says Okay add to it now take the first derivative plug in C have x - c to the first Power all over 1 factorial we don't need that one factorial it's just one does that make sense to you are you sure do one more term and see if you can hang with me plus what's the next one take two derivatives plug in C divide by 2 factorial x - c to the do you see the pattern are you all okay with that pattern for sure for real yes no yes this series do you guys see that we get a series seriously do you see that we get a a series we get a series here we get hey take function plug in C no problem take the first derivative plug in C okay no problem multiply x - c 1 take the second der plug in C divide by two factori that's where this is coming from time x - c the 2 power that's how your factorial and your power always match up you keep on going going and going and going going for Infinity this right here this is called the tailor series now it's going to be a real pain in the rear but we're going to go ahead and do the mcloren series right now okay it's a lot of work to go from the Taylor series M series are you ready to do it okay going to take us about 30 seconds here's a mclen series um if C is zero that's a muren series super hard 30 seconds but it's a harder one take about 30 there's always a cat there's no catch it's easy it's it's not easy I mean it's the same thing um listen you know how like uh every square is a rectangle but not every rectangle is a square yeah it's kind of like that okay that's analogous to what we're doing here every mclen series is inherently a tailor series every tailor series is not a mclorin series a mclorin series is simply a tailor series with C equals z you with me so if C equals z you have yourself in the cor series if Cal 0 this becomes the mcor series are they different no you do exactly the same thing it just C is zero well it makes it a little bit nicer our notation becomes nicer so F ofx would now become for mcloren series a series from Nal 0 to Infinity um it would just be this it' be the N derivative notice how the N does not become zero it's the C that becomes zero so what it becomes is uh F the derivative of FF at what zero that's a mclorin series All Over N factorial and then what happens here xus so we just have do you see how it cleans it up a little bit a little bit nicer looking because you don't have that minus C you don't have the C you just have zero because we're playing Zer it's going to look exactly the same it's going to be F of Zero no derivative for the first one plus you take a first derivative at 0 over 1 factorial * X 1st plus the second derivative at 0 over 2 factorial * x 2 plus and then so on and so forth blah blah blah blah blah blah blah blah blah okay quick show hands if you understand the notation and the ver the verbiage the idea of a mcloren series and the idea of a Taylor series and how they differ do you understand how they differ yes so on a test want to ask you for hey find the mcloren series represents this uh this function would you be able to do it well not right now but would you understand what I'm talking about yeah it's a tailor series with C equals done now here's a couple I'm not going to write these down but um couple pie of information are quite important firstly a function that has a power series representation will be represented by a tailor series it will be represented by a Taylor series the converse is not necessarily true but we kind of assume it to be true um if we find a tailor series based on the derivatives of some function that function that we get from that series usually represents our function very well we assume that to be true for this case there are cases where it doesn't but we assume it to be be true for us does that make sense so what that means is that if you got a function and you represent with the power series it's going to be a tayor series if you find some tailor series of function that you didn't know that series that you get usually represents the function if you base it on the derivatives of the function that you you're looking for that's what I'm trying to say can we please do an example to clear all this stuff up yes i' like through at least two of them the first one's really quick really nice and easy so here's how your question on a test will most likely look no matter what class you're you're you ever take in calculus this is typically how it looks so if we have a function like this and I say what I want you to do I want you to find the tailor series that represents this function at a certain point find the tailor series that represents this function at the point Cals Z can you please explain to me um another way that I could have worded this it would mean the same thing find The M bam do I have to say find the mcloren series at C equals z no no I just say find the mcloren series and that would be exactly the same thing as what I just said right here so if I say find a Taylor series that's equals zero this means find the series that's all it means you okay with that I'm going to give you the steps on how to do this if you follow these steps every time this is going to make your life a lot easier so here's the Steps step number one because our tailor series are based all on derivatives what you're looking for here and it's not that hard to actually do but it's hard to find the pattern so I said find the pattern is the hard part okay check it this part or i' rather where am I at this part this part never changes that's going to be n factorial this part never changes that's going to be x - c to the N or x to the N no problem whether you're dealing with mclen or a Tor tayor series okay all you're looking for all you're looking for is this guy or this guy that's what you're looking for does that make sense and if you have a c then it's not just going to be C it's going to be like centered at one or at or whatever you have it at it's going to be around that one point so you're going to have an act number what you're trying to find out is find a pattern that represents the N derivative of f so in order to find the pattern that represents the nth derivative of F at a certain point like zero or whatever number I give you in this case I gave you zero you need to start with the derivatives so step number one is find several derivatives of your function start with several derivatives of your function so find derivatives of F and I want you to start looking for a pattern so I'm going to do that right here and we're going to do another uh column right next to it so leave yourself some space so we start with uh start with the function itself f ofx equals what please now I think I gave you probably the easiest example in the world right now okay what's the derivative of x oh oh but wait what's the next derivative okay but wait a minute how about the next one oh do it a lot okay do like typically I do three or four derivatives uh in this case this is pretty straightforward so we're not going to do that but you usually do like three or four derivatives what you're looking for is the pattern so that you can go down to here and say all right now I know what the nth derivative of f is you see you want a pattern so that you can plug in a number to that so if you know the nth derivative of x you can find the nth derivative of C zero or one or two or whatever sometimes it's not this easy I'll give you another way to do it in a minute uh but sometimes it is so you're looking for a pattern on your derivative if you can find one so what is it pretty good guys are Cham so number one find a derivative of F and look for a pattern number two plug in xal C it should stick with the same pattern sometimes you you you will definitely find a pattern here okay it's going to be apparent to you but what we're we're not really looking for this this is X the N derivative f f f at X I'm looking for at C so our next step is plug in xal C make sure you find the pattern here this is the big one so here's what I do over here I'd find F of C I want F Prime of C I want uh F dou Prime of C only in our case we know what our c is you're always going to know what your C is what is our C here Z so I don't want just F of C I'm really looking for f of Z and frime of zero and F Prime is of Z and F triple prime of Z and ultimately this is what I want I want f as an nth derivative of zero let's see if we can do this you see what we're doing here we're trying to find a pattern based on N for this thing so what's up hey what's F of Zer when I plug in what's frime of0 what's F Prime of 0 what's F triple prime of 0 do you see a pattern here what's F what's n derivative of F at zero if it never changes you're just going to get one show hands if that makes sense to you so if you can find a legit power uh sorry legit pattern for the nth der of f ofx all you got to do is plug in zero to right here it's going to work for you does that make sense to you sometimes this is really hard so besides doing just that we try to look for a pattern amongst these terms and we try work this down and get the n f at whatever point we're trying to find as soon as you find this everything else is easier everything else becomes really really easy are you ready for it to get really really easy are you sure so here we go just plug it in so if we want to find a tailor series at C equals 0 that's in the chlorin series then what what we know is that our function is going to equal a series that looks like this it's going to be f as an N derivative of F at Z over n factorial x - c to the N okay by definition by what we just proved this is the mccorn series this is a tailor series at oh actually I'm a little wrong this is the McLaren series if I just do x to the N it's a tailor series at zero because when I do zero I'd have x - 0 to N I would have a zero for my C show P feel okay with that one that is a mcor series that's a taor series at C equals 0 they're the same thing now all we got to do is did you notice why we only have to find one little piece of information all we're looking for here is that guy that's it this stays the same that stays the same it's that guy so we find our derivatives cool find a pattern if you can sometimes you can find a pattern if you can plug in the number c find a pattern definitely the where you have to do it whatever you get as the N of at C this whatever this is It's usually this is the most basic example I can even think of it's usually based in terms of n so base this in terms of N and put it right there you catch that so all you do is a little substitution so this thing changes to you tell me what's it changed to you what's uh the N derivative of F at zero what is it it's always one over what and then perfect that's exactly right exactly exactly so step number one find derivative step number two plug in xal C step number three as soon as you have this thing set up your series now step number four is where the fun begins step number four well hey does that look like something we have dealt with before in this class could you find me the interval of convergence yes ah does that sound familiar did you just do homework on that or should you have just done homework on that yes now you go through the ratio test or the root test or something that will give you the interval of conversion you see what we've just done here is we said cool this is awesome you know what that's that function has a power series representation at equal Z that's a mcloren series well we've just found the mclorin series but you need to tell me where it's good for the where it's good for is the domain the where it's good for is the X's that make the Ser series conversion remember that a function is only represented by a power series when that power series is convergent when we actually have a sum do you remember talking about that so we're looking for the interval of convergence the domain or the X's which makes make this power series convergent that will represent that function did did you catch all that it's in the previous section if you want to R review review back on the power series idea so last little thing step number four is find interval of convergence that is all the stuff we've just done are you starting to see how this this chapter because before it's like well does this even build at all are you starting to see now how this builds why we did all that junk we did all the the convergence tests because of the end points power series we did the power series The Ratio test for the power series and the root test for the power series we did power series for tailor series and my Florin series and we did those for approximation by taor polinomial which are do in the next section so here's our last thing find the interval of convergence you will I want say almost always you'll often times use the ratio test for this so I'm going to speed through this really fast this is not the section on learning the ratio test this a section on just using it now this is not a problem anymore this is part of our problem the problem really is right here as soon as you do this you've solved the problem the rest of this old stuff so we're going to do a ratio test limit as n approaches to Infinity absolute value this would become by the way you can simplify this make this x to the N Over N factorial instead of one you know make it a little easier for yourself x to the n + 1 over n +1 factorial all over x to the N Over N factorial so far so good do you see where all that comes from yes just our ratio test tell me when the ratio test says convergence when this limit is one that's why we're doing this to find the interval of convergence so we keep on going uh that's a limit should I drop the ACT value right now no oh please don't do that very good so we have x n + 1 / n +1 factorial not a problem and then n factorial over x to the N what now simplify let's do it we're going to do it fast this is gone we have n+ one this is gone we have X that gives us a limit as n approaches Infinity of remember what I told you about this collect your n terms separate off your X term so your n terms become 1 n + one yeah that doesn't have to be an absolute value because this end's always going to be positive we start at one but the absolute value of x that is important show P be okay with that so far now be careful on this you got to be really good at this this be able to determine what this limit's going to be what is this limit going to be all the time for anything that you plug in for X remember X has to be actual number can't be Infinity for anything you plug in for X is this going to be zero yeah it's something over infinity that gives you zero times whatever I got I don't care what that is this is going to be equal to zero for all X for all X you with me now now why is it important that we got a zero here does that help us at all ah this was the ratio test since zero is always less than one ratio for all X can you tell me what my interval of convergence is if I can plug in any X and I get zero Z is always good so if I can plug in anything that says my interval of convergence is any number this series listen carefully this series converges for any X do you get that anything we just proved it right there it's always 0 less than one by rati test a converion for any x what that means is that this function is represented by rorin series for any X any X you plug in I can represent it with this series so for all X means our interval of convergence would be interval of convergence is infinity to Infinity now if you're wondering why in the world are we doing this what what's the point here's the point what's kind of cool here's what we would say by the way uh we would say that f ofx = e to X this here's the point can be represented by the series Power series N = 0 to Infinity of x n/ n factorial uh that's just right here that's what we found and then you state the interval for which this thing converges which is the interval for which our power series works for a function on the interval negative Infinity to Infinity that's the appropriate way that you write out your answer you say hey the function that we started with don't forget what we're doing here what we're doing here is we're actually representing this function with this series says this function can be represented by this series on some given interval if this had been different we would have had to check end points remember checking end points do all that jump now since it's Infinity so well since our limit is zero we get ne INF to Infinity we don't got to check all that stuff but here's the point what you can do now is use this to approximate this for instance uh understand that this what this is this is a what is that well we we actually do it if I wanted to find F of two that would be e to the second power if I if I want to be as accurate as possible I didn't need a calculator whatever here's what I could do with that it's really cool notice how I have an e here does this have an e in it no yet when I do my series if we did this if I plug in zero I would get x to the 0 so I'd get 2 the 0 over 0 factorial I would get one one plus I plug in remember my X is two in this case are you with me on this my X would be two you know what maybe it' be easier if I did this uh f ofx is 1 plus if I pluged in one I get x to the 1 plus if I plugged in two i' get x2 over 2 then X 3r over 3 factorial then X 4th over 4 factorial do I have that right can you double check me on that one make sure I'm good am I yes no okay so if I want F of 2 basically if I want e^2 well I have 1 + 2 + 2^ 2 over 2 + 3 2 to3 over 3 factorial + 2 4th over 4 factorial and if I do that forever and add them all up I will find out exactly what e the second power is is that is that interesting to you this def have an e in it but when I add them all up I get e squ you could do e to the first if you wanted to you can find out how much e is based on just this polinomial that's crazy so if you want a good approximation do it to the fifth term add them all up you're going to have a pretty darn good approximation for what e to the second is or E the first or whatever you want to plug in if I want to e to the first I just plug in one e to the seventh just plug in seven and then add them up and you get as accurate as you want to on that approximation show hands that one made sense kind of cool right that's the idea all right let's continue our examples on how to find Taylor and mcloren series now if you remember from last time I gave you some steps on how to do this I want you to read those steps to me what's the first step in finding a tailor or mcloren series what should we do first how many how many derivatives you know what as many as it takes for you to find a pattern if you can find a pattern on the first derivative fine like e to the x that was really easy uh on some of them you can't do that so do three four five however many it takes to actually find the pattern do derivatives what's the step after that very good in our case xal well Cal 1 is that's going to be our C so we're going to be plugging in one and then we find a pattern for that after that set set up series good and after we set up our series we're looking for very good it's going to be a function right we got to find out the domain of our function that is the interval for which are series converion which are power Series converion so step one let's go ahead and let's find some derivatives we always start with the function itself like Ln of X after that well we start taking first derivative second derivative everybody what's the derivative of Ln X please good now I'm going to write that differently because we're going to be looking for a pattern here so when I do 1 /x um you usually I want X to some power not 1/x so I'm going to write this instead as yeah X to the1 power so let's write this as x^ the1 that's going to make our derivatives a little bit easier as we go down the way and it's going to make us find a pattern pretty quickly are you with me yeah okay so we're going to go fast through this stuff hang on to your hats here what's a second derivative please okay I'm going to write -1 x to the -2 is that okay with you yeah and here's why we're going to do that when you're doing this when you're doing your derivatives do not do things like this when you do a third derivative I don't want you to multiply the- 1 * the -2 and get two I want you to look for a pattern yeah a negative * Nega is a positive but really we're going to get 2 * 1 x -3 does that make sense to you are you with me remember you're looking for patterns here you're looking to make a pattern out of this guy not to get the number to get a pattern let's do a fourth derivative and a fifth derivative and then we'll find it so the fourth derivative of x we write a little four up there what's going to happen it's going to be negative very good and it's going to be 3 * 2 * 1 * X4 does that make sense to you do you see what's happening had I put 1 1 2 six you would not have seen a pattern here does that make sense so when you're doing this don't don't do that don't do 1 1 2 6 20 whatever you're you're getting okay so let's do one more the fifth derivative of x we're going to get is it positive or negative so we're going to get positive and then 4 * 3 * 2 * 1 * x to the very good do you see your pattern yes tell me something about your pattern right now as far as the sign goes also do you notice something our pattern does not start with our first term this is not part of our pattern this doesn't even work with it it has no exponent at all so we're going to we can start our pattern after the first term so after that this would be like the zero term at the the next term so we don't have to start it here we notice our pattern goes okay positive negative positive negative positive negative positive negative positive negative that's going to be something it's going to be alternated do you guys see the alternate I'm talking about secondly tell me something about my exponent increases decreases by decreases by one that's right so it's negative if you want to see the absolute value increases sure so1 -23 45 tell me how it relates to the derivative that we are taking to the derivative that we're taking 33 44 5 5 the same next one will be sixa six does that make sense yes okay tell me something about the coefficient of my x's what's it doing it's a factorial that's right now is it n factorial is it the power factorial is this oh yeah it's nus one that's right so it's really close but it's still factorial but it's one less than the power we have does that make sense yeah let's check it out if I omit the negatives and positives this is 2 - 1 factorial sure is this 3 - 1 factorial is this 4 - 1 factorial and 5 - 1 factorial and 1 - 1 factorial Z factorial yes yeah 0 factorial is one okay so we could do the nth derivative right now what we know is that it's alternating yeah it's alternating it says it's going to be -1 to some power we just got to make sure it starts correctly for us so we're going to have n or n minus one up here somewhere we'll deal with that in just a second okay next up uh well we're going to have what how do I represent this okay so if I get get rid of all the negatives say I don't even care about those we're going to have an nus1 factorial for sure and then x to the N power does that make sense to you yeah you sure M okay now this part is what I'm worried about how can I represent because right here goes positive positive so let's just let's forget about this one let's look just here so I don't want positive positive negative positive that doesn't really work for us that's not our pattern so our pattern doesn't really start until our first derivative term so omit that how can I make it positive negative positive negative positive negative how can I start that off is it 1 to the N or1 to the nus1 if I start at n = 1 nus that's right and you know what if you're not good at think of man I don't really I don't know well if you can't find a pattern here that's okay what you're really looking for is Step number two plug in the numbers here's F of I'm sorry what are we plugging in even one PL one okay so F of one and fime of one and fpre of one and plug them all in for one it's got to follow the same path pattern so it's got to be alternating it's still going to have a factorial somewhere in there that's why we did this it's a look for the pattern and make sure it's still the same okay so check it out uh F of one is what is it zero zero good because if I plug in one I get zero now we just determined that's not going to be part of our pattern so we're going to be omitting that term how about this one one how about the next one how about the next one if I plug in plug in one you're going to get 2 * 1 plug in one you're going to get 3 * 2 * 1 plug in one you're going to get positive 4 * 3 * 2 * 1 do you guys see that all we're getting here is this this series of numbers that be multiplied together we don't have any more X's because one to any power is still one so basically if you just omit your X's this will get this happens to be0 factorial this happens to be netive 1 factorial this is positive 2 factorial this is negative 3 factorial this is positive 4 factorial you with me still is it still alternating yeah does it still have a factorial yeah in fact this is going to be1 to the N -1 and then n -1 factorial you can see something here as well check this out if you're good enough to find a pattern which some of you are and some of you maybe you're going to struggle with this this is kind of hard to do in in general if you follow the steps I've given you try not to get just the numbers try to find your pattern this becomes a lot easier okay but look at this if I take my my x = 1 like that's my C okay take my xal C which in this case is one and I plug it in do you see that if I plug in one here I get exactly what we found over here hello do you see that it's going to be it's got to be the same so if I plug in one if I plug in one I'm still going to get the same thing if I found my pattern correctly now if that's our n power of F at one then here's what we're looking for what we know is that a tailor series must look like this our tailor Series has to be Nal in this case um listen in this case we're not going to start the zero term are you listening we're not going to start the zero term because if I start the zero term that's this guy and that's not following our pattern so we're going to change to nals 1 you're going to see another reason why in just a minute so we're not going to start at zero we start at one because our pattern doesn't start till our nals 1 does that make sense to you not n equals 0 not the zero derivative not the function itself but at Nal 1 so 1 to Infinity we've got to look like okay the N derivative at remember Taylor series C at C uh what's our C right now okay and then say what C to the x- okay x - c very good x - one two the N over factorial perfect so feel okay with that so far so we found our derivatives we you actually found a pattern to it if you find a pattern if you find the N of F at X then all you need to do all you need to do is plug in your Center if it's zero plug in zero if it's one plug in one and you automatically get the N derivative of F at one if you simply plug in one that's literally what this means now I've given you both these steps because if you can't do this if you can't find the pattern sometimes you'll a find a pattern just from plugging in the number this is really what we're looking for anyway so if you can't find the pattern here use this to plug in your numbers and then find a pattern from these numbers show hands feel okay with with that idea once you do that understand that what we're trying to do here is do a tailor series or a mcloren if we're centered at zero tailor series are always n derivative of F at your your point C at your Center well that would be at one hey we already found that then it's xus your Center x -1 to the N power over n factorial we proved it last time so all we got to do now is what let's make a substitution so our tailor series therefore is going to be n = 1 to Infinity I know I'm going to have an X - 1 n Over N factorial can you please tell me what goes here have I lost you no what goes what goes there then perfect that's exactly right it's literally this thing if the N of f evaluated at one is this which we found in two different ways really then we're going to plug that in so-1 to the N -1 * n -1 factorial okay I really do need to make sure that you're okay with this so far um I don't think I'm going to give you any any sort of recap I've done it a couple times here derivatives find a pattern plug in numbers find a pattern this is what you're looking for the reason why you're looking for this evaluated at this point is because that is what a tailored polinomial is all about it says find your nth derivative evaluated at whatever your C is equal to we have that make your substitution these guys are always the same it's xus your Center xus your c a value at to some power n Over N factorial do you really feel okay getting that far now before you go any further simplify it if you can for instance I've got an n -1 factorial over an N I should be able to simplify that can you simplify n -1 factorial Over N back forth you're left with nus one you're not left with nus one you're left with n n factorial is n * n -1 * N - 2 * all the way down to 3 2 1 are you with me look at this what's that n- factorial so n factorial is bigger than n -1 factorial by one term by multip multiplication by n does that make sense to you so don't jump to don't jump the gun and go n minus one no this is less than that if this would have been n + one then yes n plus1 would be up here but it's not this is n -1 factorial this is n -1 factorial time n therefore this gets simplified with all this and we get 1/ n that's what we get so this goes boom boom n so our tailor series becomes 1 to Infinity of -1 to the nus1 * x -1 N All Over N that's what we really want to work with that's going to be nicer than working with this thing for us for sure are we okay with that one yes you sure okay so we've done derivatives got it step one we've done plug in the num step two we found a pattern we've made our tailor Series this is what we wanted to do this was step number three now step number four are we done with this you got to tell me where this thing converges so what's the next step interval of convergence very good how right side people how are we going to do the interval of convergence what would you use here absolutely the ratio test for sure it's got well it had a factorial uh but it's got a power of n and something that doesn't have a power of n you're not going to want to do a root test here do a ratio test also ratio test gives us a nice interval here if we know that the limit is less than one we have convergence do the ratio test for me right now please oh by the way one more reason why you can't start at n equals 0 here let me have your eyes on the board real fast if you started at n equals 0 what's wrong with this picture you couldn't even do it all right so we start at n equals 1 firstly because the pattern doesn't even start till the first derivative n equals 1 derivative and also for the definition of having a defined uh series here you can't even start at zero okay so continue your ratio test I'm going to do a board here real quick so we're going to do the ratio test but I definitely want you to do it on your own please we should be pretty much pros at doing the ratio test I want to make sure you get the same thing I got so can I have your eyes on the board here real quick how many people did the uh a subn plus1 term here and got this and over a subn term here and got show P you got exactly that cool what's the next step what should you do simp simplify as much as you can what's going to happen to my negative ones they gone so we're going to get a limit and approaches Infinity do we get rid of the absolute value ladies and gentlemen no we really don't do that when we have x's in here we are going to multiply by the reciprocal of our main denominator so here we're going to have our this piece gone still have this piece still have this this piece then we're going to well this piece is gone absolute value takes care of that for us multiply by reciprocal we have n of x -1 to the N what's the next thing that we can do here can we simplify yes we're going to simplify then we're going to collect our n terms and collect the remaining X term everything else should simp simplify here so what do we get get okay I'm going to get N I like that I'm going to get a limit n approaches Infinity of n/ n + 1 absolute I'm going to get an absolute value very good the why why can't I pull out the N Over N plus one and not have an absolute value tell me explain to me why I can do that very good n Over N plus one is always positive we start at one go to Infinity so this is always positive we collect our n terms no absolute value maybe parenthesis if you really want to be technical about that put parentheses around it uh what's going to be inside of our absolute value under one or over one where is the xus one on the numerator very good because this simplifies all of this that little + one that says that you have 1 power of x -1 more than you had here therefore x -1 is still on a numerator absolute Val is a little bit big here so we get our n/ n+1 no problem collect our n terms we simplify we get x - one still an absolute value now what do we do take a limit very good what's the limit of n/ n+ one one very good so this thing is equal to absolute value of x - one now tell me something about the test we use what test do we use what's that tell us about con converg of a series so whatever we have here must be less than one for convergence you with me that gives us our interval that's great so right now we know that this series is going to converge when the absolute value of x- 1 is less than one that's when we know this is going to converge that's when we know that this tailor series will represent our function Ln of X you with me that's what we know CU it only works if the series is going to converge that's the only time we get out put for our function which is the only time we can actually get a function here now from here can you tell me my radius of convergence what is it as soon as you find whatever is being raised to the N power right here as soon as you find that in your absolute value that right there is your radius of convergence so the r is one now let's go ahead and do the interval of convergence from here what we know by the definition of the absolute value in equality is that1 must be less than x -1 must be less than one you follow me on that one that's what a definition of absolute value inquality is now we're looking for the interval so we don't stop here we want to get X by itself so when we get X by itself we have to add one to all three of our zones practically we get zero is less than x is less than two you okay with that so notice it is one in the center of our interval right in the center it's supposed to be there is the radius still one can I add one and sub want to get these two numbers that's why we get our radius here first and then we we get our interval now here's the issue what about 0 and two ends H end points we got to check our end points check your end point actually you know what we'll do it together I want to be a little bit quicker about this let's check our end points any questions on the pattern at all so this was just to find our pattern just basically just to find this piece right there everything else we already know so here we go here's our end points what are end points right now Z so zero if my original tailor series goes from 1 to Infinity of -1 n -1 x -1 n over n and I say I want to plug what what number I'm plugging in Z and where am I substituting that so this becomes the series N = 1 to infinity1 to nus1 time what oh 1 n that's right over n you all okay with where that's coming from if you plug in 0o for X you're going to get 0us oh that's 1 1 the N here we got our first little piece 1 nus 1 we got second little piece when we plug in zero for X we get 1 n now simplify this before you try to do any tests on it the1 nus 1 * 1 the N does not look very good right now so what we're going to do is because we have these common bases being multiplied together what are you supposed to do with your exponents when you have common bases being multiplied together you add them so this becomes series N = 1 to Infinity of what over n I I'll tell you it's going to be negative 1ga 1 to the 2 N - one 2 nus one okay show P should be located with with that one now we're going consider this for a second what does 2N minus one always give you plugin ends what's it always give you odd number gives you odd numbers it's very good so we're starting one right so you plug in one we get 2 - 1 that's one then you plug in two we get 4 - 1 that's three plug in three we get 6 - 1 that's five we get odd numbers that this the expression 2 nus one this gives you odd numbers now how much is -1 to the 2 N - one then what is it always negative it's always negative very good because it's always going to be to the 1st 3rd 5ifth 7th 9th whatever always odd then -1 to the 2 nus 1 you know what guys this is just equal to 1 that's all it is does that make sense to you because it's always going to be1 to an odd number negative 1 to an odd number is always going to give you negative 1 therefore this series becomes how much is this whole piece do you understand why yes if this is always odds this whole piece becomes -1 n you sure you're okay with it now do you remember that we can pull constants out of our series negative sum from 1 to Infinity of 1/ n oh my gosh we finally have it tell me something about the sum from 1 to Infinity of 1 over n what is that Harmon har that's harmonic or it's a p series with P equals 1 either way you say this what happens to this series I don't care about the negative what happens to this series negative Divergence is still Divergence so this thing diverges so harmonic therefore Divergent tell me something about the end point of zero is that going to be included in my interval of convergence no therefore is that going to be in the domain of my function as is represented by my power my tailor series no no it's not going to be my my domain okay the next one next end point we got to check is the two so let's plug in the two if we plug in two this series goes from 1 to Infinity of -1 to the N minus one if I plug in two how much do I get one yeah 2 Min - one gives you 1 to the N All Over N now this one's going to be much more straightforward not going to have any fancy pants math to do up here okay how much is one to the N that's easier I guess this wasn't too hard anyway you got to know that 2 N - one is odd numbers so this becomes -1 to the nus1 All Over N this whole thing is one one to the N is always just one 1 time anything gives you the anything hey tell me about this series what do you know about it alter it's alternating it's alternating harmonic harmonic what what do you know about the alternating harmonic series could you do a different test to show that if you forgot it do alternating series test you got it don't do absolute convergence because this does not converge absolutely if you do ab absolute convergence you get harmonic and that says Divergence says not absolutely convergent it's conditionally convergent by the alternating series test so this is the alternating harmonic series by the alternating series test this thing converges you state all these things you tell me what it is you tell me what you're using to find out convergence or Divergence say harmonic ay Divergent no problem alternating harmonic series I know that's converion already or by alternating series test it converges so here's what we know we know that the function FX = lnx can be represented by the Taylor series and then you show the tailor series you say it's -1 to the nus1 x -1 to the N All Over N factorial you you show me what the tailor series is you say hey look at this this is what we're doing here folks don't lose track of what we're doing don't just miss the forest for the trees and go through this process and don't understand what we're doing what we're saying is this you ready this function is represented by the Taylor Series right here centered around one on the interval and then you tell me what the domain of this function is question do we need the uh nus one factorial sorry it should be n just it's this you want simplified the simplified form so let me make sure I got it right we got N1 n -1 x -1 N All Over N yeah give me this give me the simplified form this is the easiest one okay you're not want to do this if you can do this and get the same numbers so do this so our function can be represented by our tailor series we got it right here but only on the given interval that you just found this the only time that this series will converge therefore it's the only time that this series will represent this function it's on what interval parentheses Z bracket good so parentheses because zero does not included bracket because two is that means you can plug in any number from zero non-inclusive up to two inclusive and what you plug in is going to represent that function if you do the series to Infinity show F if you understand that one okay cool tell you what I'm going to show you a couple other problems uh as many as I can do to get you really good at this because there's some other techniques that you're going to have to learn here in order to be successful at doing your Taylor series and the cor series okay last chance for any questions you have any I want to start fresh do you have any questions on on this one have I explained it well enough for you all to understand it cool so our next example write down F ofx = sin x yes we're going to do trigonometric okay series don't for your pants it's fine and what I want to do right now I want to find the mcor series what's that mean find the Tor series that's right mclen series is a tailor series with C equals to zero not a problem so what we're looking for ultimately is this what we want excuse me what we know is we want a series that has the N der at zero x to the n Over N factorial does that make sense to you hello yes no that's a tailor Series right it just so happens that c would be zero making it a mcloren series that's that's what our goal is is to find that thing now where do we start say what now F the nth derivative of F this is hard to say the nth der of f that's what we want so we start doing derivatives that's step number one so D okay we'll start with our function our function sinx what's our first derivative please what's our second oh come on we I know that more than one of you know this what's our third derivative what's our fourth derivative is there a need to go any further no okay I hope not s cos cos cos cos cosine here's the problem oops what's this what is that cuz I don't know because it depends on where you stop doesn't it it depends on whether you stop on a remainder of when you divide your your derivative by four if you divide it by four and it's zero cool 1 2 3 0 1 2 3 0 it's going to be a cyclical pattern it's like number Theory okay so yeah you you don't know what that's going to be so this one we can't do it this way sometimes we can't find an N of f explicitly because it's going to depend on what remainder of four that we're in on our cycle well you can even see it clear here number two step number two says okay start plugging in oh I'm sorry what number we supposed to plug in here Z very good our c is zero and the first derivative at zero and the second derivative at zero and so forth and so on let's see if you can give this a try what's uh the function itself self at zero yep what's the first Der at zero and then and then and then so we go this is weird 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 what is the n derivative at zero again I don't know it's one of these three choices it's either Z 1 or Nega 1 but I don't know what it is we're going to have to go a different way around this all right so when you come up this and you can't find a pattern start listing out the terms of your mclen series or your Taylor series start listing them out so we're going to start there so if we really list this out arm mclorin series uh from yeah we'll do over here aror series what's our very first term well this would say that you start zero right you'd have X to 0 that's 1 over 0 factorial that's one uh You' take 0 derivatives that's basically just F of zero remember doing this like the first day we did our MCL and our our Taylor series we started listing out the terms for it you'd start off with f of zero did you follow me Taylor would be uh F of C no problem then we go okay cool uh we add to it the first derivative at zero time x to the 1st all over one factorial that would simply be one we okay with that one then we'd have the second derivative at zero we would have X2 over 2 factorial no problem then we'd have the third derivative at zero we'd have X to 3 all over 3 factorial then we'd have the fourth derivative at 0 x 4th power all over 4 factorial and we would not stop I want to show up hands if you can get from here to here you understand what I'm doing show hands if you do that okay so AR yes no AR mclorin series says when you plug in zero you get no derivatives bam no derivatives at zero x to the 0 is 1 0 factorial is 1 then you plug in 1 so our first Nal Z term Nal 1 term plug in one you get a first derivative no problem x to the 1st all over one factorial plug in now two you get a second derivative you get X the 2 all over 2 factorial now plug in three you get a third derivative x 3 all over 3 factorial you plug in four fourth derivative of 0 x 4th over 4 factorial are you paying attention does it make sense to you let's plug in the numbers what's F of z z what's fime of 0 1 1 * X is X does that make sense watch carefully what's the second derivative at zero this is why we did this so this whole thing becomes zero what's the third derivative what's the fourth derivative zero zero do you guys see what I'm talking are are you with me at all I don't want to lose you here but I want you to I want you to be with me but I got to move too so do you get that the the function at Z zero yes no first derivative at 0 is 1 second der at 0 is z third derivative at 0 is negative 1 fourth derivative at 0 is 0 * anything is zero plus how about the fifth derivative is it going to be zero or one or negative 1 it's going to be one 0 1 0 1 0 1 0a 1 0 1 0a 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 it's like a song having fun right this is it's cool well now let's simplify this a little bit do I really need to include all my zeros now we come to a fine point that maybe it makes a difference maybe it doesn't I like to think that it does okay we're going to come to a fine point this really is the first term second term third term fourth term fifth term we're going to redefine our series here because what's going to happen is if I omit my first and my third my zero withth and my second term and my when n equals three and or two and four I'm going to get x - x Cub over 3 factorial + x 5th over 5 factorial - x 7 over 7 factorial plus blah blah blah blah blah blah blah forever and ever and ever and ever can I need quick show hands if that makes sense to you now if I try to Define this as a series in terms of n my NS are going to compete for instance my n's are stemming from here aren't they are you with me which means that this is going to be my zero 1 second third fourth fifth 6th sth but this I want to consider my first second third four so can I define a series in terms of n yes but understand it's a different n it's a different Series so what I'm going to do right now I'm going to use the letter K so we have no ambiguities of what's going on okay we're going to Define this thing right here as a different series so tell me on this series from K = 1 to Infinity tell me what's going on tell me something about that that you guys see alter it's Alterna okay that's easy let's start with that1 to the is the first term positive or is the first term oh you know what let's start at zero actually start at zero just to keep our our uh MCL Series in mind here start at zero first term positive first term negative so the only way that we're going to do that is if we do oh which one k n minus one would give us zero and that'd be negative 1 K just K just be K because if I do negative 1 to the zero I get posi 1 does that make sense if I do negative 1 to the first I get negative second I get positive third I get this is going to work for us that's where we're getting our alternating from is 1 to the K are you sure you're okay with a okay a okay you get an A because you're not a problem now multiply tell me what our X's are doing what are the powers of X doing odds they're odds hey we just dealt with odds we just dealt with odds well let's see if that works um if I plug in zero wait a minute I'm gonna get Negative I'm gonna is this x to the negative one power how can I change that to make it work that'll work that'll work so if I do zero okay what's 2 * 0 plus one let's do uh plug in one now plug in one so remember zero should give me my very first technical term if I plug in one it should give me this term are you with me plug in two it should give me this term that's why we redefined our K so we don't get ambiguous with that so plug in your plug in one two * 1 is plus one hey plug in let's plug in two 2 * two is four plus one five boom now what's the denominator doing it's exactly the same but with a factorial you better make sure this is accurate before we go any further okay so we start off derivatives are really easy pattern is really easy but there's no way for us to explicitly Define it because we don't know what this is going to be the N der F I don't know it's supposed to be a little n by the way I don't know what that's going to be so we write out the terms mclen series says you know what you have F of 0 frime of 0 * x frime f Prime of 0 x to the 2 over 2 factorial x triple prime of 0 x 3r 3 factorial and so forth and so on except with our pattern that we just found we know we're going to go coefficients wise 0 1 0 1 0 1 0 1 that's going to basically omit half our terms they're all gone so we just take the terms that actually have something and what that does is that takes a a series with like two n terms and gives us a series with n terms or practically we have a series with n terms and we get a series with n / two terms or K terms where k equal n / 2 well if that's the case let's redefine it so we don't get confused on the the terms themselves we say okay let's let's start k equals to zero and go to one and go to Infinity then we have this is this what's that say what's this part say for your series alter it's alternating so this says we're alternating no problem this says we're taking x to the odd Powers over odd numbers factorial this is our mclen series now what we going to do with our mcloren series say it again ratio test definitely ratio test it's got a factorial in it right we're not done here you see what we've done is we've defined we know that F ofx equal sign can be defined by this mclen series it can be defined by it but we got to have the interval of convergence so do the ratio test notice word K so your limit should go from k k approaching not n but K approaches Infinity so so go forward this one should be quite easy for us the simplification is going to be a little bit crazy but too too [Applause] hard oh my gosh1 to the k + 1 * x to the you know what if we do k + 1 here do you see it's going to be 2 * k + 1 + 1 do you see that 2 * k + 1 + 1 okay and this is going to be same thing 2 k + 1 + 1 factorial sounds like we have a stutter doesn't it plus one plus one no no no1 to the k x 2 k + 1 all over 2 k + 1 back forth no problem but I want to make sure you guys have exact the same thing that I have here show hands if you do have exactly that thing is the ratio test getting pretty easy for you yes it should be right we've done it like 50 million times what's going to happen to my Nega 1 cancel I mean don't matter absolute value takes care of our 1 any power now of course we have a complex fraction we'll write this as a limit don't change your K to an N absolute value don't lose that on our main numerator let's distribute notice this you're going to have 2K + 2 + 1 that's 2K + 3 you guys get that one yeah X to 2 k + 3 all over 2 k + 3 factorial here we're going to get 2 k + 1 factorial and then here x to the 2K + one yall still all right with that one just some algebra we multiply by the reciprocal what should you do now what should we do now simplify as much as you can now be careful with this can you simplify X 2K + 3 and x to the 2K + 1 I want to say yeah I want to say yeah too so if this is X 2K + 1 this is X 2K + 1 + 2 do you see the plus two in that this is x 2 k + 1 + 2 that + 2 is really just an x² do you get that yes then you get the whole thing okay now this one's a little weird we got factorials we've done this only one time before if this is 2K + 1 factorial this is 2K + 3 factorial think about this for a second 2K + 3 factorial means listen carefully I'm not going to write it again I've already done it this means 2K + 3 time 2K + 2 * 2K + 1 * everything below it so what when you simplify this this 's completely gone you with me this one you have the terms until you subtract down to 2K + 1 so you have 2K + 3 no problem you have the term 2K + 2 no problem the next term would be 2K + 1 would not yes that's going to simplify term by term with this one so our limit therefore becomes the limit as k approaches Infinity what do we got 1 2 K do we need an absolute value here no you actually don't you have an X squ don't you or positive these are all going to be positive just make sure you put them in the right place we're going to have a one over what's going to be on our denominator 2 2 time X2 see where X squ is coming from 2K + 3 2K + 1 now let's do this some there tell me how much is this limit always going to be equal to as K approaches Infinity always for any X remember you tell me the X first right you say x is blank I say okay pick a k that goes to infinity and now we take one divided Infinity that's Z 0 times any number that you've already told me is zero and this is oh wait uh why is the zero important tell me in relationship with the r test why that's important lesson one says that it converges right now does it matter the X for which we plug in to get our zero no we're going to get zero no matter what x we plug in so this thing equals zero for all X that means Z is less than one for all X can you tell me my radius of convergence convergence is infinity what's my Center does it really matter if we going to Infinity no not really but the center is zero what's my interval of convergence so here's what we say and you write this out FX = sin x can be represented by the series N = 1 sorry zero my bad 0 to Infinity of I'm going to use this series okay but I'm going to change it back to NS just so we're we're our format okay -1 to the N it was a dummy it's a dummy variable it's a dummy K it doesn't really matter what we use dummy plus 2 n + 1 factorial on the interval negative Infinity to Infinity as we just just showed hang on for one second I know I'm going to go a couple minutes over here but I want to show you one more thing it's quite interesting uh quick show of hands if feel okay with this so far now this is pretty pretty freaking rad okay which means if you want to find out sign of I don't care what you want to plug in s of three sign of pot sign pot or two I don't care what you want to plug in all you got to do is plug it in here this is remember this is the series right this is where we got this is the same thing this is the same exact thing so you want to you want to find s of 3 S of -2 sign of 1. 1846 I don't care plug it in here and here and here and here and here and figured out for the first however many terms you want to and you can be as accurate as you want that's what the power series will do now the last thing very very fast this is interesting remember me telling you something about how we can take take derivatives and integrals of power series remember doing that we did it in the in the very end of our last section well if FX = cosine X and cine X is the Der of sin x and sin x is x - x 3 3 factorial plus X 5th 5 factorial - x 7th 7 factorial plus blah blah blah blah check it out do you agree that sin x can be represented by this on any interval we just proved it just showed it then if I take a derivative of sign take a take a derivative of all this crap what I get hey what's the derivative of sign I'm sorry what's the derivative sign so if sign's represented by this stuff if I take a derivative of sign and I take a derivative of this stuff the derivative of s is cosine cosine can be represented by the derivative of this stuff which means one minus x 2 over 2 factorial look at what happened bring down the three the three is going to cross out with this three and give me 2 factorial plus x 4 over 4 factorial - x 6 over 6 factorial plus do you see what happens it just lowers everything by one power and one factorial specifically we get a mcloren series of it's still alternating it's still alternating it's still X to a power but it's only even Powers you don't get the plus one over 2N factorial do you guys get that so we didn't have to do any work for it we took a derivative of sign derivative sign's cosine no problem take a derivative of term by term basically look at this this is really this is really cool um check this out could you have done it right from here take a derivative this goes here cross out with 2 n + 1 you get 2 N subtract one from my exponent we get this last little bit what's the interval do you remember me telling you that when you take derivatives and integrals your interval of convergence does not change only the end points since we don't have any end points our interval is exactly the same so cine X can be represented by this thing on that interval sure if F feel okay with with that one okay okay so getting to our last example on how to do a tailor or mclorin series from scratch from scratch means that we invent our own series after this what we're going to be doing is using known Taylor mclen series to fit our function to it you see that's a lot easier and so a lot of times we can manipulate what we have into what we've already made does that make sense to you so the last thing we're actually going to make um and what it is it's it's a polom 1 + X2 some power K where K is a real number so K could be like 1/2 K could be negative K could be positive like two or three in which we would get like a binomial expansion that's actually what we're going to get anyway so I'll I'll show that to you right now so once we start do you remember the first step when we start any tailor or M series what do we do good so we're going to find a series of derivatives here we always start with the function itself F ofx and we find our first derivative first derivative is going to be well if you notice this it's going to be the chain rule over and over and over and over again so practically we bring down the K we get 1 + x the chain rule gives us one and then we get K very good that's right and we keep on going the second derivative would give us K * K -1 * 1 + x to the K - 2 are you hanging on with me here okay the third derivative would give us K and then it would give us K minus one and then would give us K - 2 and then it would be 1 + x and then it would be K - 3 you still okay yeah we're just doing a general power rule or a chain rule whatever you want to call that we'll do one more a fourth derivative would be tell me very good now we want to make the jump how about the nth derivative of f the nth derivative let's look at this for a second think about what's going on what's going on is that I'm multiplying but it's not Qui K factorial what's happening is I'm doing I'm doing k then I'm doing K minus one I'm doing K - 2 I'm doing Kus 3 but I don't get to the K minus n do you see it I get to the K - n + 1 the K minus one more before that n does that make sense to you okay also I'm doing my K well this is to the K minus n so let's check it out first derivative minus one second derivative - 2 third dtive - 3 fourth derivative -4 nth derivative K minus let's let's wrap it up first dtive K -1 second Dera k 2 third dtive K - 3 fourth dtive K - 4 n k minus perfect that's exactly right now all the stuff I got to rewrite this I'm going need more room so n derivative 1 + x to the you just said it K minus n and are you guys all right with that one yeah now here we'd have K we'd have Kus one we'd have all the way down to K minus what's this number in relation to my n like nus one it is n minus one that's right but check this out we'd have K minus nus1 do you guys see what I'm talking about here this would be k - oh N - 1 K - oh 3 - 1 oh uh K - 2 - 1 K where it go k minus 1 - one just K so if I had Kus nus1 the N minus one's got to be in parentheses do you understand what I'm talking about yeah well if I am to distribute this and make it a little nicer this is K minus n + one Kus n plus one let's see if it works we got our K sure we got our K let's see how this works is this K - 2 + 1 yeah - 2 would be well -2 then plus one gives us Min - one here would be 3 3 - 2 plus one got it and then we'd have -4 + one that's three I think I said that one backwards said Min - three and then plus one that's - two -4 + one that's three - n + 1 and that's where we would stop should feel okay with with that so far now we actually have a pattern for our nth derivative of f which means if I want to find what I what I want here to do here is find the Corin Series so if I want to find them the chlorin series what I want is that n derivative of F at zero the mclen series says it's centered at zero you with me on this one well let's do that what number am I going to plug in and where am I going to plug it in Z so that's what it says right here so if I plug in Z for X what's uh what's 1 plus 0 one one one to any power one one so this whole thing is going to become one essentially I'm just going to get this piece back again do you see it yeah so K and then K K -1 and then all the way down to K - n + 1 okay showan you located with that so far guys over here yes no y all right so if we're looking for the mclen series here's what we're saying we're saying the mclen series should represent a function f ofx in this in this case 1 + x to the K by doing this it should be a series which starts at zero and ends at Infinity it should be the N derivative at zero it should be over n factorial * x to the N if that's a MLL series does that look familiar to you yeah that's our mcloren series all right cool well hey you know what this part is going to be there no matter what it's centered at zero yeah that's why we have mcloren Series this says over n factorial not a problem this one do we know our expression for the nth derivative of F at Z in fact we just found it it's right there so now we say okay this series is n = 0 to Infinity we've got oh my gosh k k -1 K - 2 all the way down to K - n + 1 all over n factorial * X to n goodness gracious me that's a lot of stuff okay what I want to know is by show hands you feel okay getting that far do you do you see where where it's coming from do you see where where all our derivatives are coming from yes no why we stop at n minus sorry K - n +1 you see where that's coming from all right well if we plug in zero for X which is our center it's the cor series says C C equals 0 no problem just plug that in so at zero we get this thing's gone we get just this piece by definition in the chorin series is the n derivative of F at zero hey that's right here times x to the N okay that's right here all over n factorial so we have the N der of F at Z we got x to the N we got n factorial hopefully it's clear for you it's just a mcloren series you guys with me on it now let's do do the first few terms of this to see what it really looks like because many you probably going how in the world am I going to do that series what in the world does that look like here's what it looks like this series looks like this so really you could represent 1 + x to the K by doing these terms the first term is going to be one the first term is going to be one because remember what the mclen series represents are you listening mclen series represents f of0 first F of 0 says I just plug in Z I would get one 1 + 0 is 1 to any power gives you one that would be my first term then I would get the first derivative at zero the first derivative at zero would give me K but then I'd have my X for my power series I KX all over one factorial because I'd be at my first term are you guys all right with that so far so this practically speaking just represents these guys right here then we multiply by X to whatever our n is and divide by n factorial so here we go okay uh at zero term I'd get F of0 that's 1 * x to the 0th power that's one / 0 factorial that's one so we get one the next term says okay cool well we get K * X to 1st over 1 factorial that's KX then we would get K * K -1 * X to 2 all over 2 factorial are you starting to see the pattern at work now yes no you sure okay then we keep going we get K we get K minus one what's the next term actually factors would would be and then what x to thir all over perfect that's what this thing looks like you go okay cool it starts at one because the mcor series says you start at the function evaluated at your Center are you listening you start at your function evaluated your Center in a mcloren series that Center is zero so you start at F of zero that would be one to I don't care what it is one to any power is one we start at one and multiplied by x to the zeroth over Z factorial that would be one over one that's giving you one then you do the first derivative evaluated at your Center the first derivative evaluate your Center is 1 * K so it's K time x to the 1st all over 1 factorial that just gives you K * X does that make sense then you do your second derivative evaluate at your Center x = 0 okay well that's 1 * K okay so we have K * K -1 hey time x 2 over 2 factorial that's where these things are coming from that's where they're coming from the next one would be K * K -1 * K - 2 * K - 3 x to 4 over 4 factorial and You' continue forever does that make sense to you what this is called is the binomial series binomial two terms being expanded so this is the binomial series and we can make this series look a little bit different uh what this actually gives you here these things give you the binomial coefficients do you know what the binomial coefficients are have you ever heard of Pascal's triangle have you took me from calculus one you have heard of that Pascal's triangle goes 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 have you ever seen that before yes that's what these numbers are one and then well whatever whatever K you'd have it would depend on that but that's what the binomial coefficients are so this thing this gives you this one k this gives you the binomial coefficients for whatever K you actually have isn't that kind of interesting it's going to give you a line of Pascal's triangle what does Pascal's triangle always start with one one and then it depends it's gonna either do well one two and then one three one oh look at that whatever your power would be that's weird that's kind of cool so if you have a power of the fifth I'd be down here 1 five it'd be 1 5 10 10 5 1 that's interesting so this would be 1 five this would give me 10 this would give me 10 five and then one that would be my binomial coefficients so the way that we write that actually since these only represent your binomial coefficients we can write it this way this series is represented by K N this stands for binomial coefficients like that so what I want you guys to know about this thing is that this idea this whole thing from here to here they all they give you are the terms of Pascal's triangle those are the binomial coefficients this is represented by that just to make it look nicer so if you ever seen that before that's what that is quick head not if you're okay with with this so far would you like an example to see how this actually works in practicality okay before we do this I want to give you a couple notes here's your notes if K do you have this written down because I'm going to erace this for some room so these are binomial coefficients these are binomial coefficients these are binomial coefficients that's it if K is a positive integer if give me an example of a positive integer please four good what else one and any posi number 100 not any positive number no 1.3 is not a positive integer integers of positive whole number very good a positive whole number is a well whole numbers are inherently positive so a positive integer are whole numbers 1 2 3 4 5 6 7 8 9 10 not zero does that make sense to you so if K is one of those if K is a positive integer then this series will converge for all values of x okay so let me let me put this in uh in other words for you okay let's say that your function was 1 + x to the 3r is that a POS is three a positive integer then this series is going to represent this function function for any x that you put in any X does that make sense to you if if this was uh if the K was now4 is this series going to represent that for any x no but it will represent for a different set of numbers okay so here's here's the other choice I guess if K is not a positive integer if K is not a positive integer this series will converge but it's only only going to converge if x is between 1 and 1 oops more specifically if K is between1 and 0 we get to include the end point posi 1 if k is greater than or equal to zero we get to include both end points if not we don't include any end points and that would be by the ratio test so let me go through it one time as far as all this conversion stuff goes you ready I'll give you an example in just a minute we'll go through it pretty quickly but we'll give you an example here so binomial series no problem those are binomial coefficients what this series will do this series will converge and represent this function if K is a positive integer anywhere anywhere 1 2 3 4 5 6 7 8 if K is sorry if K is 1 2 3 4 5 6 7 8 9 10 11 12 forever no problem then this series will converge for any value of x that you want to put in there did do you grasp that your interval of convergence is infinity Infinity not a problem it will always converge now if K is not a positive integer that would be 1/2 -4 -3 whatever it happens to be this series will converge it will represent your function but only on the interval of1 to 1 xal 1 to1 those values of X will make the series converge if K is not a positive integ have I explained that well enough for you now there's more specifics to it if k happens to be between - 1 and 0 you get to include the end point of this interval one if K happens to be between well not between sorry positive so greater than or equal to zero then you include both of those end points that's the idea here show fans feel okay with that one now let's do an example here real quick [Music] um does this fit our binomial series that we want I think so does this look like this something to the K power yeah how what's our K okay good now can we represent this with a with a Taylor series or mclen Series yeah yes we can and tell me something will the series that we get converge they all converge where will they converge for all X or between 1 and 1 and okay now let's think carefully we're so we're not here right no K is not a positive integer K is positive is K between 1 and 0 no is K between 1 and 0 no is K bigger than zero we're here okay if we weren't here and K was like negative let's say k was -2 we'd be here let's say that K was -2 be here does that make sense to you okay so here's the deal we can represent 1 + x to the 12 Power by listing out these terms these binomial terms so what we' do is we'd say all right let's just follow this for right now so we're going to start with what good one we always start with one that was the easy one okay let's continue what's the next term what's our K so we would have 12 X Plus what's our K again so we have 12 * 12 - 1 * x^2 all over so 12 * 12 - 1 * x^2 all 2 factorial then we have what one half very good keep going could you do the next terms if I asked you to yes very good now ultimately what we're going to get down to is we're going to get this we're going to get down to 12 12 - 1 12 - to 12us n + 1 remember that term k - n + 1 okay and we would have x to the N All Over N factorial that's where we would end now we're going to simplify this a lot a lot a lot so we got to look at what these things are actually doing these fractions what we're going to do is we're going to take these fractions simplify them and look how that affects our problem are you with me you sure you're with me okay let's think about 12 times whatever this happens to be what's this going to give you okay remember what we do with patterns what we do with patterns is we don't really multiply things I know it's 14 but what I want you to see is that this is really -1/ 2 * 2 two do you see it 2^ s this one would be 12 * 12 time 3 am I right or what it be positive or negative positive and you would get three over 2 are you with me yes okay let's put this together then so what this becomes we'd start with one we'd have 12x but then the next one check this out this is where the pattern actually starts so positive POS what's the next term going to become we're going to have X2 all over 2 factorial * 2 squar this right here is -1 over2 2 if I take this this goes on my denominator this negative changes that to a minus I have an x squ on my numerator I have a two factorial on my denominator and I have a 2^ squ on my denominator the algebra can be a little tricky if you're not super good at algebra do you see what I'm talking about here yeah are you sure have I lost you I don't want to lose you okay let's go to the next term come on stick with me here folks uh do I have a plus or a minus plus it's going to be a plus because it's positive I'm going to have an X to the3 on my numerator do you see why now this was 3 over 2 Cub so what I get is where would the three be the three would be on the numerator where would the two cubed be 3 factorial * 2 3 does that make sense to you okay now the next one let's see about the next one do you suppose the next one's going to be positive or negative okay on the on the numerator do you understand I'm going to have an x to the 4th yes yes do you understand that I'm going to have a four factorial somewhere yeah yeah do you understand I'm going to get a 2 to the 4th yeah now these numbers are interesting this goes one three see where the one's coming from five three five actually it's 1 time three isn't it this is 1 * 3 * 5 the next one would be plus 1 * 3 * 5 * 7 x 5th over 5 factorial 2 5th and that's the pattern we're looking for the last term the last term as we end this it's going to be there's actually no really good way to write this pattern it's 1 * 3 * 5 you got to write that out and then go okay well where's it ending it's ending at 2nus one it's ending at an odd number 2 nus one I'm sorry 2 nus 3 because Theus one would give us two much bigger one bigger - 2 X to the n all over let's see if it matches this is our n matches our factorial n factorial 2 to the N now because this pattern starts at the second term we're going to Define it this way so we know we can represent this function 1 + X2 by doing this you guys see what I mean about the pattern starts at the second term this pattern doesn't really start right here this is positive positive and then we go negative the pattern starts here so you can define a series like this or a function like this Define it like okay well we got 1 + 12x plus and then we're going to start with our series the series does this it's starts at the if this is the zeroth term and this is the first term our n would be starting at the two to Infinity we got to start with look at this do you I see that I have the same one I the same one2 and then I have a minus I got to start with a negative term so what we're going to do is say I want this negative one to start by being negative how can I do itus one I can do n minus one um we we we're going to use n + one just so we don't have any problems with it uh we can do n minus I think n one will work as well so n plus1 did you see that's going to be the same exact thing netive positive okay cool after that well we have man the pattern starts the pattern does this omit these first two terms that's what this does says hey put the first two terms then start your Series this is like where we start our Series start your series with a negative then make it a positive then a negative then a positive that's what this is doing starting with a negative then doing positive then negative then positive are you with me still so far yeah the numerator of your series does this the numerator goes one 13 135 1357 13579 blah blah blah blah multiplying by all the previous odd numbers it's not a factorial because you're not multiplying every single number you're multiply only the odd numbers the only way we represent that is by showing it then we do 1 * 3 * 5 and then you say you're going to stop at well what we want to stop at is let's see this would be it compared to compared to This n this would be 2 * 5 minus what to get this 2 * 5 because we want to Bas on N right and it's got to be odd so it's 2 * something - one or- 3 - 5 well to get this number it's 2 * 5 which is 10 3 not one it's not stopping at 9 it's stopping at seven one odd number before that so we're going to stop at 2 nus 3 that's our numerator our denominator is only n factorial no problem 2 the N got it and then we have x to the n and that's about the nastiest freaking thing that you've probably ever seen yeah yeah crazy do we have to check for convergence yes no no no cuz we already did it I already proved it to you in the very first example we did today so what we say is that so F ofx can be represented that's what this means F ofx can be represented by this series on the interval what's our interval to one including both why why including both it'ser good yeah X is the K the K is greater than zero um it's not betweena 1 and zero also this would be all real numbers if K had been what exactly right so we'd say this we'd say that our function can be represented by this really nasty series we know it's going to be convergent because we already proved y we already showed that if K is positive integer for all real numbers X if if K is between uh1 and 0 then it's for x = -1 to 1 but only including the end point one if it's outside of that range if it's less than one less than negative 1 then it's only for negative 1 to one not including any end points for us we say hey look at that our K is not a positive integer but is positive if our K is not a positive integer but is positive we include both of our end points it goes from 1 to one that was a crazy example show hands if you hung with me you're okay on it it's a little weird right is it going to take some maybe thinking about later maybe maybe watch this part again now what we're going to do from here on out I'm going to show you how to make your given function fit a series we've already done I'm going to list out some series most of them we've done already the last two well there's two or three of them that we haven't are there any questions no thank goodness okay that's great okay I need you to write this quickly we've already done they are in your notes I'm just recapping here so here's some common mclen series that's smaller I think I all but these three we've worked with so what I'm going to do for you is I'm going to give you what these series look like I'm going to give you the the actual series notation for it um and I'll show you how to work with them because they're they're it's going to be really nice I promise you're going to like this better than what we just did well that that's not saying much uh anyway so here we go what this series is do you remember doing that it's one of the first ones we did we took an integral of this it became Ln of 1 - x and then we divided and we got this uh we got well that would represent the Ln but this series is represented by 1 + x + X2 because what it really is is simply a geometric series remember that a was 1 R was X bam this is our geometric series and it was simply represented by this and the interval of convergence was -1 to 1 edx was a nice one it was 1 + x + x^2 over 2 factorial + x 3r over 3 factorial plus and that's the series from n = 0 to Infinity of X the N Over N factorial and that converged for all real numbers sign we actually did do you guys remember doing sign remember doing some of these ones we worked through them this is all of our examples we made them from scratch sign was this it was x - x 3r over 3 factorial + x 5th 5 factorial minus and that created the series for us from n = 0 to Infinity -1 it was alternating obviously x to the odd 2 n + 1 over 2N +1 factorial and it converged for all real numbers X does that one look familiar to you yeah we did it yesterday we did oh perfect so it looks familiar uh cosine cosine was really easy cosine was just the derivative of sign 1 - x 2 over 2 factorial plus X 4th over 4 factorial minus blah blah blah blah we said hey it's still Al alternating it's now x to the even numbers over even numbers factorial and it converged from negative Infinity to Infinity oh that's last thing we did that's right okay Ln of 1 plus X Ln of 1 plus X we have x - x^2 / 2 + x x 3r 3 - x 4 4 I don't remember if we done this one or not this is the series from n = 0 to Infinity -1 to nus1 X over N All Over N and you can see that right here it's kind of a nice one this converged on ne1 to 1 positive 1 was included one is not included you know I'm probably move the last two over there um just to be clearer sin inverse of X we would have X we would have no we have not done this one okay I'm just giving this to you um so you have it we actually do X Cub over 2 * 3 we would do 1 * 3 x 5 over 2 * 4 * 5 and what this ends up being is a series which is quite interesting we start at zero we go to Infinity we have 2 to the N factorial X 2N + 1 all over 2 n n factorial squared times odd numbers these ones I got to tan inverse of X is this x - x Cub over 3 plus x x 5th over 5us X 7th over 7 or in other words that's an easy one looks nice at least we get alternating we get x to the odds over odds do you see how similar tangent inverse is to S inverse it's really similar except that's a factorial really similar though it's interesting and tan inverse only converges on 1 to one inclusive so does s inverse okay last one we just did it if you have 1 + x k this thing is equal to the series we just showed it Nal 0 to Infinity I'm not going to go through it again uh binomial coefficients time x the n and it converges on1 to 1 conver 1 to one if K is and there's special cases okay if K is positive integer it's all real numbers if K is between zero and uh1 then we get U one included if we get above zero then we get both end points included so there's special cases there now here's the point the point let me make sure I got all these right too I don't want to I definitely don't want to give you the wrong stuff here I pretty quickly I think you forgot coine no I didn't forget it just that looks good perfect got it that one's so nasty that's good Perfect all right now here's the whole point the point that I'm going to make here is that you can manipulate functions that you're given to fit these forms and if you can do that then you can you don't have to reinvent the wheel basically you don't have to start from scratch on these problems have you written this down cuz I'm going to erase it here real quick because I need to start here do you have it yeah this one you already have so just worry about about that one can I eras it yes no okay good for example let's say I give you this function on your test which I'm going to do and I say I want you to find the Taylor series at this point so I want Taylor at x = 2 or C = 2 we go for it now here's how to diagnose the problem if you're going to have a tailor polinomial oh not sorry I'm skipping ahead you're going to have a tailor series at at the point xal 2 or Cal 2 what that means is that somewhere in your series you are going to have x- 2 does that make sense to you no I can tell by the blank looks in your faces that you're like what if you're going to have xal 2 or C = 2 whatever I want to say there then you're going to have x - 2 somewhere in your tailor series do you remember your tailor series says x - c yes you're going to have x - 2 here if your C is equal to two so F be okay with with that one okay here's what you need to do whatever your Center is if you have xus 2 X plus whatever uh sorry c 2 or c equal whatever get x minus that number in your given function you must make this up here somewhere so what that is what that means is that I've got to manipulate this such that I find this in here are you listening I'll show you if you if you don't what in the world you talking about I'll show you okay you got to manipulate this to get my x - 2 so here we go from 1X 1 over 1 + x well i' think about it 1 over 1 + I would do uh x - 2 but then I have to undo it + 2 can you follow that one I'll do it one more time I have this I need this so I if I centered at two I'm going to have to have x- 2 somewhere in my I have to have that so my 1 /x plus whatever I'm going to say all right cool well 1 + x no I need an xus 2 that's what I need does that make sense to you I can't just arbitrarily subtract two from my problem though if I subtract two I'm going to have to add it back it still has to equal this thing does this still equal that yes sure it does now let's simplify 1 3 + x - 2 quick show of hands you feel okay with that so far do you see that we have the same function it's just we've made this piece in it a it's going to be a tailor series it has to have xus 2 up there somewhere that's our Center got to find your Center okay you got to be centered here now take this and actually look at this one what does this most closely match out of all of these does it match s inverse no no it doesn't does it match e to x no does it match 1 over 1 - x that's the closest that's what we're going to try to fit it so we need to make this thing fit this thing now when I say fit this thing we need to make it fit 1 over 1us I'm going to put X here in quotations x x stands for some quantity it can be a really nasty looking thing but it stands for X like whatever's there and then whatever's there we're going to take it and put it here and here and here and here and here we're going to get that thing out of it let me show you so if I need to make this fit 1 over 1 - x I need to manipulate this so that this thing becomes a one can I do that how do I do that if I factor out a 1/3 if I factor out of 1/3 I would get 1/3 * 1 over 1 + x 2 sorry x - 2 over 3 can you do the algebra on that to make sure that that's okay with you yeah distribute the three here's three here's x - 2 do you guys follow yeah now this is very close this is one over one plus something some bunch of crap okay I want 1 over one minus I got to change it to a minus it's got to fit this so I'll have my 1/3 I'll have my 1/ 1us but then this how can I make that how can I make a plus equal a minus something you can't dox + 2 explain me why you can't dox + 2 boom you have to have that does that make sense so you got to have that as your Center that's got to be there it's okay it's basic algebra it really is basic algebra but you feel okay with it is this thing still this thing yeah just nasty looking now here's the cool part this is okay this is where the rubber hits the road all right this is this is what's awesome what this really is is it's yeah it's 1/3 but inside here do you see it's 1 over 1 minus X it's just our X our junk is this junk that's our junk are you sure you're okay with it do I need a 10-second recap to go go over it or are you you sure you're okay yes some of you don't look okay make this here because that's our Center X if it's two you got to have xus 2 put that up there yeah you got to add two to it simplify it make it fit one of these things in order to do that we need 1 over 1us junk X is your junk okay if I need 1 over 1 minus junk right now I don't have it make this a oneide three got it make that a minus split off a negative now I got it now it's one over one minus junk cool cool take this thing fit it to this stuff and then see what happens so if we see what happens here we're going to fit it to this this formula what's our formula start with yep the next thing we'd have in our formula is what notice how X is the junk this is the junk you sure you okay with that yes so 1 plus we'd have x - 2 over three so far so good are you sure you're okay I I want don't want to leave you behind here because this is very important for us to know we have one we got one we got jump here's junk put your junk next thing we do junk Square yeah junk Square what's what's in between there okay plus I don't like to say the whole thing so yeah junk Square minus 2 over 3 whole thing S Plus let's do the next one and then we'll stop what's the next one plus junk Cub plus blah blah blah blah blah we're going to go forever and ever and ever okay I need to show up hands if you think that you can make it from here to there show fans if you can do that it's not that bad right you should put the whatever your X is this is the important part this is the hard part making it fit after you make it fit just put this in here it's 1 1 - x it's 1 over 1 - x put it here and here and here and here and blah blah blah blah blah blah blah blah blah and then see what happens with your series now there's one thing that we're forgetting what are we forgetting the 1/3 is multiplied by all the stuff don't worry about that right now forgotten about that don't worry about that right now wrong on the test yeah don't do that we're going to use one of these later so okay let's keep on going let's take this let's simplify it so if I take that remember oh man remember we are looking for patterns here we're looking for patterns so what I'm not going to do I'm not going to go 9 and 27 what I'm going to do is I'm going to keep the I'm going to keep the 1/3 I'm going to do a one I'm going to do a minus so notice how that's a minus we got x - 2 over three what's the next thing we're going to have a plus or a minus we're going to have x - 2^ 2 over 3^ 2 yes no we're going to have a minus x - 2 3r over 3 3r then it's going to be a plus and then so forth and so on um can you make it down that far with me yeah you see where we're getting the x - 2 to the 1st and to the second to the third yeah yeah we get to the 1 and to the second to the thir we're doing positive oh sorry negative and then positive and then negative and then positive and we have three to the first and second and third and fourth now we're looking for a pattern here so check it out let's fit this to our pattern this is 3 to the what power no this is 3 to the first this is 3 to the aha that's going to give us our one so we start with one x - 2 the 1st x - 2 the second and third we got 3 to 0 1 2 3 can and we go ahead and make a series out of this in series notation sure here's how you do it the 1/3 start starts up front and you multiply make this your series our Series starts at n equals good zero it goes to Infinity tell me something about this series that we know Alterna alter alternating and starts with a positive so we're going to have neg1 to the N cuz that's going to start positive the next one would be negative the next one would be positive does that make sense to you yeah we're going to have x - 2 to the very good all over 3 to the N perfect that's exactly you guys are getting it yeah this is 3 to the N hey 0 1 2 3 here's x - 2 the n 1 2 3 here's starting positive and going alternating no problem there's one more thing we got to do it's very easy take the 1/3 and move it inside so if you have 1/3 * this do you understand this is 3 to the 1st yep if you have 3 to the 1 * 3 to the N you get 3 to the n + 1 now wrap it all up FX = 1 1 + x can be represented by this on the interval oh wait a minute for got all about the interval didn't we do we have to check the interval no no it's already been done for you which is why we use these things no ratio test no endpoint test no nothing look at look at the board which series did we use what's the interval of convergence same thing here pretty cool right you can make them fit this is one of the harder ones that you're going to do um now we're almost done with it almost I I kind of lied a little bit this would work this would work if your X was just X but keep in mind don't stick with me here for just a second okay this would work if your X was just X but your x that you used here is your junk okay so we have one more thing to do remember that your X your X Works between -1 and 1 do you follow yeah okay just put your junk into where this junk is so what's your junk negative okay Sox - 2 over 3 that's your jump multiply by three oh wait a minute what's negative due to inequalities F add your two flip it back around we get the interval 5 take your function make your Center fit after you make your Center fit manipulate so it fits one of your series you already know after that use it your X whatever your your X is up here is whatever your junk is here whatever you made it fit as okay so this right here this becomes your X just don't forget that at the very end when you get all this stuff done when you find your pattern when you make your pattern work it's not going to be that hard when you make your pattern work take your junk put it back into your interval and then solve for it solve for what that interval is does that make sense to you yeah okay now I'm going to pause the tape here real quick uh if you need to go you may go I'm going to stay here for another 30 minutes or so and lecture uh I'm going to put this on video tape so if you want to just do this later you can um but so if you want to if you need to go go I don't want to make you any later than you are for whatever you want so I'm going to give you about 30 seconds if you want to head out head out you're not going to offend me okay if you want to stay I'm going to be lecturing I think you're going to enjoy it if you do stay but if not have a great spring break okay 30 seconds go or stay this doesn't look like something I can afford to miss you can watch it it's going to be the same it's not the same it's not the same it's not oh I don't want to do this to you guys if you really need to go oh no it's fine okay all right so now we're going to continue that's all you're saying did this make sense to you did you grasp that you go okay take your junk plug it in no problem just don't forget about your interval your interval yeah it's going to be based on this but this says that your X your stuff is going to be between 1 and one solve for your actual X and you get here between 1 and 5 now we're going to continue we have um we have about two two and a half more examples they're going to go quicker now that we have this idea uh so what I'm going to do is I'm going to put up the example we're going to talk about what series we're going to base it on I'm going to show you the things that we can do and we'll end so next [Applause] example what I want to do with this I want to find an theoren series you'll notice that in the previous example we based it on a mclen series didn't we because I gave you common mclen series if you need a tailor series you have to fit the xus the center in there not a problem if you want a mcloren series it's even easier so if I have x^2 sin of 2x well what series are we going to base that on sin x so what we know is that uh if sin x is is equal to x - XB 3 factorial + x 5 5 factorial minus blah blah blah blah if that's the case look at what we can do it's so nice all we got to do all we got to do well is this exactly like this is this exactly like this no exactly like this no this say sin x this say sin 2x don't fret don't worry if sin x equals this then sin 2X equals plug it in 2x - 2x 3r over 3 factorial + 2x 5th over 5 factorial minus and you keep going like that notice how here uh I keep using the word junk I'm going to still your your X your junk is now 2x so you just plugged it in now fortunately for us because we have a mclorin series you don't have to fit in xus anything it be zero it would just be X does that make sense to you go cool all right well your junk is 2x just put it in here we got 2x we' have 2X Cub we'd have 2X to 5th should feel right with with that one all right now oh man is this the same as that no no but it's really close so if that's not the same uh what what else could we do say what now how do I make this look like this m you can actually do that right now if you want I'm going to show you how to do it later uh it's it's really easy so I I prefer doing it this way but you could multiply by X2 everywhere you could do that it's going to come out exactly the same it's just the patterns a little harder to find so we're going to start finding this pattern right now so uh working on the pattern let's see what it actually does what this is is this be 2x no problem this would be remember I'm looking for patterns here so I'm going to have 2 to the 5th no I'm not I'm have 2 3r X 3r over 3 factorial I'm going to have 2 to 5th x 5 over 5 factorial I'm splitting this up for a reason the reason why I'm splitting them up is because I'm going to be multiplying by X2 later and so I'm going to have to M split them up at some point or another I'm going to split them up right now show fans if that makes sense to you now let's make our series we're about good to go on this so our Series starts at zero it goes to Infinity uh tell me something about this series that you know Alterna yeah it's alternating and it starts off with a positive so that would be n that's right tell me something about else about this series OD numbers it's odd numbers that's right it's taking two two odd numbers it's taking X to odd numbers all over odd numbers factorial we have to use a plus one here for odd numbers because we're starting at zero do you guys see what I'm talking about if we start at zero and we get a one and a one this says well at your zeroth term you're actually getting one so 2 * 0 + 1 gives us 1 2 * 0 + 1 gives us 1 all over 1 factorial so we're starting it says zero but we have to make them odds but we got to start at one so it's not minus one this time it's plus one you see y that L you you're does that make sense okay so here's what we know we know that this thing s of 2x would equal that series with no problems now how do I get from s of 2x to what I want you can do things like that so if I multiply by X2 and I multiply by X2 listen it's an equation okay it's going to work this thing rep is represented by this series therefore X2 * this thing is represented by this series times x^2 therefore x^2 * sin of 2x is equal to our series we go from 0 to infinity1 1 the n 2 2 n + 1 what's that going to be 2 2 + 3 2 N yeah do you guys remember that you're adding exponent so 2 n + 1 + 2 that's 2 n + 3 now I'm going to stop for just just a brief moment do you understand that um you could have multiplied by X2 here and got the same thing it's just your twos would be different than your X's so your powers of two x's is this going to change to 2 N Plus three now we're almost done with this one too this they're going to go quicker after this uh look back at your notes and tell me the interval of convergence for our sign not sign inverse for our sign they infin to Infinity what that means is that it our s is convergent on Nega Infinity to Infinity our s of 2x is also convergent on negative Infinity to infinity and even when I multiply by x^ s it's still convergent on negative Infinity to Infinity so what we say we wrap this thing up and we would say that uh FX = x^2 sin of 2x can be represented by our nasty series on negative Infinity to infinity and if you forgot what that means here's what it means it means this if I want to find out this as evaluated any number I could actually do it with a series I could plug in my give me a number any number three three plug in three if I wanted to plug in three I could actually represent it by plugging in three to this series working all out to infinity and adding all up together and it will work for any value what it says is that this series will converge for any value of x therefore for any value of x this series this power series Taylor series mclen series will represent our given function show F if that one made sense to you so can you man manipulate what you have to get what you what you need sure multiply this by two you get this take find your pattern find your series multiply by x² no problem just a couple more the reason why you don't have a whole bunch of other uh mcloren series why you only have a a set amount of them for me is because a lot of times you can manipulate what you're given to what you want to get remember CCH the definition for CCH no we don't uh it's a do you remember what it means hyper that's what the H is very at least we understand that okay so uh well if you don't remember it here's what this meant this meant e to x - e to X all over 2 do you remember that one yes okay cool well if you split this up then what we really have here is a combination of two different series which is really cool so think about this um look back at your notes and you tell me what e to the x equals as far as your series is X N you know what that's great but we're going to have to do it term by term because we're going to have to find this pattern because we're going to be combining two series okay so tell me term by term what e to X is represented as 1 plus say it again keep going plus X over 2 factorial and Patrick's right that would be X the N Over N factorial quick head now if you're okay with with that so far awesome now if we can fit these things to this thing this thing to this thing and then subtract them when we just fine so check this out this is kind of interesting so if I do oh wait a wait a minute one e xus e thex well if I have one look at that do you see that e to X is just going to be this bunch of junk so this is going to be 12 * 1 + x + x^2 2 factorial + x 3 3 factorial plus blah blah blah BL blah minus 12 * here's our 1 12 here's our e tox you tell me does this fit exactly what this is what's our junk here so negative X is what I'm plugging in so if I plug Inga X then I'm going to get one I'm going to get please look carefully if I'm plugging INX I'm going to get- X do you see what I'm going to get plus X2 2 factorial do you see why I'm getting plus because if I'm pluging a Negative X I go okay One onega X cool what's x^2 postive x okay what's my next One X to 3 is still X it's X 3r over 3 factorial plus blah blah blah blah are you sure you're okay with this now something really really cool is going to happen when we do this check this out ignore the 1 halfs for a second because naturally if I'm subtracting something multipli by 1/2 I can subtract the terms themselves and multiply by 1/2 later did you catch that so watch what's one minus one those are gone do you see it y now this would be x - x so that does not simplify CU that's going to be X Plus X that would give us 2x but what's junk this yeah junk X2 2 factorial minus x 2 factorial do you see that every even power is gone every even power so the squared squared the fourth with the fourth let's combine what's left over in fact if I just factor out my 1/2 we're doing some Advanced outs we're here all kind of in our head if I just factor out the 1/2 what I'm going to end up getting is well here's and here's remember minus and negative is a plus we'd get 2 x Cub over 3 factorial Plus 2 x 5th over 5 factorial plus blah blah blah blah blah blah blah did you catch that we we did some Advanced algebra here we're Distributing our negative and we're combining our like terms basically we're factoring under 1/2 no problem our negative would distribute and then we're combine like terms if you get that your ones are gone your two power twos are gone your power zeros are gone power twos are gone power fours are gone power every even power is gone your odd Powers will be added together x + x 2 X X3 X3 2x3 X5 2 that's 2x5 now when I distribute my 1/2 what happens with this one and that one and that one and so on and so forth so this becomes x + x 3r over 3 factorial + x 5th over 5 factorial which makes a series oh let's do this together where does it start starts at zero goes to Infinity tell me about my powers of X what do I have odd Powers remember if I'm starting at zero is this odd power going to be 2 N minus one or 2 n + one plus one because I have to start the one so yeah you're right x 2 n + 1 all over tell what I get on my denominators beautiful that's exactly right so FX equal cin X can be represented by this on now look back uh what what mlin series did we use to do our work with e x so look at the interval of convergence for e x therefore if we used a a m series which has a convergence of negative Infinity to Infinity then this series also has convergence of negative Infinity to Infinity so when we get something that's not explic listed we don't want to make this thing up on our own okay we don't want to start doing derivatives of this it's crazy we don't want to do that we want to base on something we know already the definition is this well that's 1/2 e x oh look at that this is exactly what this is this is just this stuff with a Negative X plugged in everywhere we get here we get here multiply by 1/2 no problem factor out the 1/2 distribute your negative lots of stuff simplifies out and we got a very simple series to work with where we already know the conversion so when I say represented that means that this series will give us this function for any number X you plug in and that's kind of cool we have one more thing to do and then we're going to be done okay this is the last thing this is uh this is going to blow some of your minds right now okay this is what I really wanted to get to today uh because it's so interesting and so cool I want to lead you down this road here and I want to consider this here's my question do the interal that wasn't a question was it do the can you do it yes how do you do it do the interal do the interal can you do the integral is a UB going to work no heck no it's not a definite integral so I'm not going from anywhere to anywhere can I do the integral itself oh yeah here's how firstly well you know that uh e to the X we actually just work with it 1 + x + x^2 2 factorial + x 3 3 factorial plus blah blah blah blah right hey can you make this thing look like that thing what's your junk here so plug in your junk if you plug in your junk I still have one then I have 2 no not - 2x^2 it's going to be a minus it's just going to be X2 are you looking at this here's our X right take this that is now your X does that make sense to you this is your junk plus we'd have x^2 2ar over 2 factorial plus x^2 Cub over 3 factorial plus blah blah blah blah blah well that's 1 - x^2 + x 4 / 2 factorial - x 6 over 3 factorial plus what would the next one be come on go for it are you with me on that one y are you sure you're with me let's write this as a series our series would be 0 to Infinity tell me something about our series what would we get say what definitely alternating and starting positive so1 to the end we'd have x to the tell me something about our X's what's going on all evens that would be 2 N no pluses no minuses just 2 N when we plugg in zero would get out one we plugged in one we get out two all over what that's right in factorial that's brilliant exactly right show hands feel okay with this so far you're sure you're okay on getting from here actually from doing the X to Thea x s you see where that's coming from we're just taking X squ and plugging in every single place we got it now this is so cool if we want to do the integral if we want to do the integral well let's just since we know that we about go since we know that this equals this since we know that this equal you believe that right and it converges for all real numbers since this converion is all real numbers this is for all real numbers well if this equals this and I want to find the integral do you remember anything about calculus you can integrate on both sides of an equation well then all I need to do is integrate this side and integrate this side integrate this side and integrate this side what that means is that if I do my integral I could do it from here actually I'd have x - XB over 3 plus X 5th [Music] over 2 factorial * 5 well you know what there's an easy way to do it I showed you this before if I integrate this one what's going to happen to my power remember doing the the calculus on these things so this is a series negative 1 is still going to alternate I'm going to have x to the 2 and what do you do with integrals do you add or you subtract I'm still going to have an N factorial but I'm now going to divide by my 2 n + 1 you can't put them together there're still an N factorial but now you're just dividing by that new power that's exactly would you what you would get if you did this ter term by term term by term would give you x - x Cub over 3 + x 5th over 5 * 2 factorial - x 7th over 7 * 3 factorial it would give you alternating still odd X do you see the odd X's odd powers of x 1 3 5 7 no problem it's giv you over odd numbers 1 3 5 7 times your n factorial 0 1 2 3 factorial so you can do integrals of functions that you could not do any other way besides this is that not interesting to you isn't that kind of cool we've done an integral of this thing that integral will be represented by this series if I want to find the integral of whatever I whatever I want it's going to be on the series from Infinity to Infinity because integrating series do not change the interval of convergence we talked about that before do we need a plus c for that or no yes yeah I don't even know if I put it on my notes but plus plus c means you're completed that's right did that one make sense to you are there any questions on this before we end so so you got you got that you got this from here to here right you got that if this thing equals this I can integrate both sides and if I integrate both sides well if I have a series I can integrate the series if I don't have the series I can integrate term by term and it's going to come up with exactly the same series that's really cool Michael are we going to have to do that in this class yeah have fun we just did