Transcript for:
Understanding Power Series and Convergence

okay so moving on to where the rubber hits the road why we dealt with the series in the past few sections is because we want to talk about power series and what these things are so real quick brief explanation of what a power series is what a power series is is a series with some variable in it not like n but like X like our typical variables that's being raised to some power that's why we call it a power series because we have a variable raised to some power let me let me uh write that down and explain what they would look like so a power series it's simply a series with a variable X that is being raised to some power and that power is based on N now what we're going to talk about in this class are two types of power series we're going to talk about power series centered at zero and we're going to talk about power series centered at some constant C so there's two types we talk about um other than the fact they're centered different places there's really not any difference between them so there's two types of power series Power series series number one power series number one looks like this it's a and this is going to look very interesting to you because you've never seen it before it's a series we start them at zero so the first term starts at n equals 0 typically and what it's going to look like is well that would be just a normal series wouldn't it just a subn something like uh I don't know n^2 over NB + 1 that's a that's a type of a subn or or whatever we we've had many of them Ln over of n whatever we had a lot of different types of sequences from where we get our series now for a power series the difference between what we've done before and what we've done what we're doing now is instead of just having this we also have a variable raised to some power that's what's giving this this idea of a power series it's now have has a variable in it does it make it harder or easier just different it's it's going to use a lot of the same Concepts that you already know we're still going to be doing things like the ratio test and absolute conversion and all sorts of that stuff we had to have that and you're going to see why as we go through this section okay so when we get down to checking for end points when we get down to radius of convergence when we get down to finding out whether these are convergent or not we're using the same tests okay it's all the same stuff so we had to have all that so that we could accomplish this section so let's talk about what this would look like just the first couple terms how it's going to look what where do we start these uh these power Series so if I plugged in zero well I don't know what this is explicitly but I know it's going to be a sub Z true yeah and it would be x to the 0 what's x to the 0 so our first term is simply going to be a sub Z you with me on that one yeah we're going okay plus it's not alternating typically unless we have something in our a subn that makes it alternate plus I would plug in one and I would get x to the good and then I would plug in two and I would get x to the second I'd plug in three and get x to the 3 and so on and so forth forever and ever and ever and ever because this thing stops at Infinity you guys okay with that so far yeah now just a little heads up I want you to think about functions do functions ever stop no functions going into so right now what we're doing here is we're actually defining a function which is interesting I will'll talk about that in a minute so number two the other type of power series you can have is one that's centered oh you know what I forgot to tell you a little something I mentioned it but I want to write it down for you to make sure you have it what this is this is a power Series in the variable X which is centered at the origin the origin would be our zero I said centered at zero centered at origin you can use them interchangeably this is centered at zero or centered at the origin so this is a power series in X which is our variable this means it's a power series based on the variable X which is centered at the origin or zero why is it centered at zero or centered at the origin why is it centered there I want you to look at the x is anything being added or subtracted from the X so zero is being added or subtracted from the X nothing's being added or subtracted therefore it's centered at the origin or centered at zero quick head on if you're okay with that one compare to this suppose I had a series starting at zero never ending a subn x - c to the N of course the first few terms are going to look really similar to this let's check that out and then we'll consider the differences between our our series themselves if if I plugged in zero I'd still get a sub 0 I'd still get x - c but this is going to be raised to the 0 power what's anything raised to the 0 power so it starts out exactly the same do you see y mhm plus I'd still have a sub one but this time instead of having x to the 1st I'm going to have x - c to the 1st and instead of having a sub 2 * x 2 I'm going to have x - c to the 2 other than that there's not any change to it it's just where this thing is centered so that little C that says xus C that c is what your Center is so if I have minus three we're centered at three if I have plus two we're centered at2 did you catch that are you sure because it's minus C so minus negative would make that plus for us so anyway uh we have a power Series in X still but this is centered at C where C is it it's a constant now of course I'm going to give you some examples in a little while and we'll talk much more about this uh as we go through but for right now we're not going to work with these things I just want to give you examples of how they look would you guys like a couple examples of how these things look real quick please okay so here's an example just for a few terms we'll see what these things are doing and of course we'll talk a lot about them as we move forward you know the first thing I want to know I want to know if you can tell that this is going to be a power series just by looking at it can you tell it's going to be a power series just by looking at it yes what's it got there that none of our series before is yeah it's a variable being raised to power that's a power series now quick question do you think this is going to be centered at zero or centered at some other number centered at the origin or not what do you think does it look like number one or number two looks definitely like number one because we have nothing being added or subtracted to our X itself if it's not centered at the origin it's going to be x minus or X plus something now you have you understand that okay also can can you pick out my a subn from here can you pick out my a subn what is it just cover that up and that's a subn that's it so this is the the sequence right and then after that we're getting our series from and then after every term we're just tacking on an x x to the 0 x to the 1st x to the 2 x that's what's given us our power Series so really if we wanted to do this we could write out just the sequence here and put x to the 0 x to the 1st x to the 2 x the 3D after every single those terms and we'd have the appropriate series now of course we're going to do this all at one time but let's let's try it just for the first few terms uh where do I start again Z let's plug in zero what's netive 1 to the zero one so we're going to have one what's x to the 0o one and what's Z factorial one one Z factorial is defined to be one if you didn't know that it's one so our first term here is one you okay with that so far next term let's plug in one if we plug in one what's 1 to the one so I know I'm going to be subtracting hey what type of a power series is this Alterna Alterna yeah that's right so how about uh plug in one here we get x to the over over what over one over one factorial you can put the one factorial here if you just want to see the pattern okay so right now we just have X but if you put one factorial you're going to see the pattern in just a minute next one is it going to be plus or everybody sure and what we'll get is X over good minus X 3r over 3 factorial plus and do you see the pattern it's easy to see the pattern once you get the first few terms that right there is the first few terms of our power series let me give you the next one I'll make a couple SATs about this I do want you to understand that this thing is centered at the origin there though have I made that very clear for you so it's a power series no problem cuz you got X to a variable you notice how the variable here is 1 2 3 blah blah blah blah blah it's going to continue continue we have it centered at zero because nothing's being added or subtracted from our X now compared to this next one what's that going to do for us okay okay it might look a little nasty but don't worry about that right now we're just going to kind of talk about this one for a second um first thing you notice what type of a series is this is this a power series what tells it's a power series variable it's a variable it's being raised to a power now let's see is this more like the number one or more like the number two so is this centered at the origin where is it centered very good do you understand why it's not centered at negative 4 xus that's right it's x minus the center just like finding a root like if you did xus 2 the root of a a polinomial xus 2 means your x intercept is at two so that positive two you can think about like always a changing sign well it's not really it's defined to be x minus your Center so the center is Pi / 2 if I had had X oh my God sorry I said Pi 2 I meant Pi 4 if I had X+ Pi 4 the center would be at very good exactly right so let's figure out a few terms here what's the first thing we plug in everybody zero so if we plug in zero we're going to get let's see here here's one plug in zero here oh wait so am I going to start with n oh that's that's going to be different than we had before so this actually does something here so don't just automatically assume we start at one all the time that doesn't work so this does give us one this gives us well 2 * 0 is 0 + 1 is 1 so we're going to start with x - 4 if I plug in zero here I get one factorial which is just one you guys okay with that so far y next term is it going to be positive or negative plus or minus this is an alternating series because that thing is alternating so we're going to get minus I'm still going to get notice notice what happens here and what doesn't happen here actually are there any ends in here at all so is this thing ever going to change no no it's not changing so you're going to be writing things like X or x - four over and over and over again the thing that changes is the power it's a power series the power is changing of our variable and this denominator so I still have x -4 but my power is going to change and what and my denominator is going to change what's my power going to be remember that I plugged in zero first I'm going to plug in one now what is it so this is 2 the3 over and notice how this is the same as this so if this is three this is three factorial this is five this is 5 factorial 3 factorial plus we have x - < / 4 to the 5th over 5 factorial what it's giving us is odd numbers the next one would be 7 over 7 factorial 9 over 9 factorial does that make sense to you that's our power series centered at Pi 4 now we're going to talk about the the theory kind of some some notes behind this first thing do you notice that when I'm doing this when we're doing this we're actually getting a function out of it do you see that M we get a function based on X so a function of X these power series are all going to create functions of x so if these things create a function of X then what we can do all the time with our power series do you understand that what I'm talking about a function of X yes no if I actually do the series there's no more ends the end does 0 1 2 3 4 5 to Infinity what I get is a a series of terms all of which have variables in them and that variable is always X does that make sense it's a function of X so what that means is that I can define a function of x to be equal to my power series um by the way I'm wondering if you understand why I'm using this form of our power series rather than this one do you understand that this form is like the general form of this if I have C equals 0 then it gives me this one so therefore I can always use this and be correct as far as my general power series quick head now if you're okay with that one okay so does that make sense to you if this thing is a function of X we already said it was because what happens is I'm going to get a subn is going to be a number a constant number for every individual term and when I put zero or one or two or three or four or five or whatever into my n that's going to be a constant this of course is going to be a variable that's going to be a con constant so I'm going to get a function in terms of whatever my variable here is it's going to be a function in terms of X now this is what's going to be interesting please listen carefully if I have a function equal to my series oops when I plug in some X when I plug in some X I'm going to get a series of terms correct that series is either going to converge or it's not not going to converge does that make sense to you so I'm going to have a series I'll have x's in it you say hey give me an X whatever you give me I'm going to plug in my X either the series will converge or the series will not converge does that make sense if the series converges we would get a sum out of it does that make sense if the series does not converge of course we wouldn't be able to find the sum so if the series converges we get a sum that sum is the output of my function the input of my function is of course my X but I only will get a valid output if my input makes my series convergent I'm going go through that one more time because this is kind of the theoretical wait a minute does that make sense to me let me say it again okay here's what's happening when I plug a number in it's going to create a series so plug a number in for X not N N's already zero to Infinity I'm plugging a number in for x here and here and here and here and whatever I'm going to get a series that either converges or diverges does that make sense to you if it converges I get a sum that sum is the output of my function if it does not converge diverges then I don't have a valid sum okay it'd be like infinity or it would be undefined we can't do that so what we're saying here is this function is defined with a domain remember that domains all the things we plug in with a domain of all X's such that the series converges this is a function with a domain of all X's such that the series converges if the series does not converge then I don't have an output for my function does that make sense do you need me to explain that one I tried to explain it three different ways but I they all end up sounding kind of the same because it's the same idea all three all three different ways it's the same thing are there any questions on this idea I know we haven't done any examples but I want to make sure this is clear before we go absolutely any further to make sure that you get it do you get it that uh this thing can be defined as a function of X because X is our variables do you understand that series either converge or diverge okay do you understand that for certain values of x X this series will converge and for certain values of X this series will diverge some of the times so sometimes you plug in anything it's going to converge sometimes you plug in anything it's going to diverge but a lot of the times we have some x's that's going to make the series converge and some x's that's going to make the series diverge do you get that do you understand that if the series diverges I can't really represent this very well with the function because I'm not going to have a sum the sum is my output I sound sound very s my output uh some is my out therefore when you put in an X that gives me then I when y'all put in an X I say when I was 13 my parents took me on a trip around the United States and we spent a long time in the South and so I actually did pick up an accent from down there in Louisiana and Texas and all those places so by the time I got to Boston I sounded all messed up you know that age you're picking up a lot of L so anyway anyhow so if this thing is a function of X and I plug in certain numbers of X some will make it convergent some will make the series Divergent those values of X which make this series convergent mean that I get a sum for those values of x if a series a conversion I get a sum so for the values of X which make this thing convergent I get an output of my function therefore we have a function with a domain of of all X's that make my series convergent the X's that make the series Divergent we don't care about because we can't define the function there show hands if that made sense okay you understand a power series then that is what a power series is now let me give you a couple examples on how we actually do this and what we can do with it not not any hardcore examples yet we're still kind of on the the note taking part of this so we got the power series idea power series idea does make a function so that our domain is all the X's that make the series converge that's it now let's look at what we can do with it let's start with the most basic power series I can think of it's just this the series of n = 1 to Infinity sorry from one 0 to Infinity of x to n firstly is it a power series yes okay uh what's my as subn one one it's one yeah that's a very simple power series but still a power series let's look at some of you guys zone out don't zone out on me here uh let's look at the first few terms just to get an idea about what this is what's the first term here next term next term good and then it continues continues continues forever never never here's what's kind of interesting these power series are going to represent series a lot of the times that you've seen before or that you can work with in the same manner you work with the other series for instance this right here might look weird to you this is a geometric series that's all it is it's just a geometric series and I'll show you why this is simply a geometric series do you see it yet no no let me let me put this on the board do you understand that a geometric series is a * R to the N yes that's a geometric series uh well if that's a geometric series by the way um sorry when does this thing con do you remember when it converges R it's all based on the r when R is absolute I see the value greater or equal p series greater than one make sure it's on your note card note card note card it right now yeah by the way um that R I don't know if I ever told you this that R is called the common ratio did I ever tell you that no it's called the common ratio like 20 sections later whatever so um a geometric series converges when the common ratio R is in absolute terms of absolute value is less than one that's what this this thing is now I want you to consider this for a second so the series X to the N has a common ratio of x oh look up here does this fit this format yes what's our a no the a is one a would equal one what's taking the place of our what's taking the place of our R the X that's why I said has a common ratio of X that is the r in this case so so what's really interesting is that this thing is really a geometric series now the question is when do geometric series converge when R when the r is less okay so so I I don't want to lose you here do you understand the idea I'm trying to make this look like geometric series it is geometric series it's just the a is one the r is x in this case that's geometric now if geometric series converge when the absolute value of R is less than one then this series converges when the absolute value of x is not zero less than one less than one less than one that's the idea of the common ratio the common ratio well okay it says converges if the common ratio is less than one this has a common ratio of X it will converge if x is less than one in terms of absolute value or if you want to think about it this way when -1 is less than x Which is less than one so is this diver it's convergent let's talk about it one more time just to make sure that you are with me on this thing first things first are you starting to see that this is geometric it is this it just happens that R is no longer a constant it happens to be a variable now what we know about geometric series is that geometric series converge every time this number is between1 and one therefore I don't give a heck what that's not even it's a thing now I don't give a crap what uh give a what this is in India what this is as long as this number is between negative 1 and one non-inclusive does that make sense to you that's what this says I don't care what this number is as long it's between 1 and one non-inclusive this will converge this would if I give you negative 1/2 is that going to converge how about uh 1/3 1/4 1/5th 1 16 and then forever until you get to zero how about zero how about 1/2 how about 1/3 how about 1/4 all the way to yeah it's going to converge for everything as long as this is between as long as this is between because this is that it is a geometric series does that make sense to you you sure so our domain is right here remember that the whole thing hey this thing that defines a function of X doesn't it are are you are you catching this picture this defines a function of X when will this function of X conge it will converge when X is between 1 and 1 that's what we're saying here this is a function on a very specific domain is it a function defined everywhere folks everywhere no it's defined for a very specific set of x's what's the set of x's that this is defined to be that's our domain that's it now here's what's also very cool do you know how to find the sum of geometric series a r so in our case here the sum is a over 1 - r a over 1 - R check this out so if F ofx and this is this is kind of how we'd write it mathematically okay we said a lot of I said a lot of words here you didn't say very much forget you guys but I said a lot of words all right here's what this says this says I have a function this is a function of X this function X is geometric if it's geometric it will have a sum the sum is defined to be a um how much is our a one over 1 - R what is our R aha the common ratio is r or in this case X put it all together folks do you understand that's a function of X do you understand that's geometric it will have a sum M do you understand that this sum is only valid when this thing converges do you get that and when does this thing conge yeah this is awesome this is really cool because uh CU here's what this says do you do you guys this is this is equal right and this is equal right because this a series is a sum and this is the sum of any geometric series when X is between Nea 1 and one because that's when this series converges that's therefore when we can find the sum what that means is that this series actually represents the function FX = 1 1 - x that's what it represents but only on a c interval only between what two numbers you got it I'm using interval notation here 1 to one the interval 1 to one um so if you want what in the world are you freaking talking about L here's a deal uh give me a number between1 and one a half like a half so if you wanted to you could plug a half in here right and find out what it is if you wanted to you could plug one2 in here and here and here and here forever and add it all up it's going to equal exactly what you get out right here cool it's kind of weird right so we're using these series to represent functions or rather we're using functions to represent series same idea soens if you understand the idea power series represent functions that's what they do because they have x's in them functions of X they represent these functions only on certain intervals for which the series converge if you get that idea you do understand power series do you understand power series my speech is al off power series said parachute parachute you understand parachutes jump out of planes if they open you live how it works usually okay uh show hands if you honestly do you feel okay with that one are there any other questions whatsoever on that so so our X is our input our sum is our output that's how it works couple other little notes for you before we get on to I do want to kind of go quickly through this because I want to do an example to give you the idea of what you're going to be doing here question oh that's just a proof right what you to show this or take we find out if that thing actually converges it AES or not well that's the whole point is that your X is a variable correct yeah so I can't tell you this converges all the time unless it converges for all X's what you're doing is you're saying okay if this is this and this is a geometric series then I know when it will converge what you're basically doing is giving me the interval for which this will converge that's what you're doing not the other way around you're telling me that it will converge or won't converge it will at some point give me the interval that it does converge at does that make sense then the interval that it does converge at is is the domain of your function otherwise there would be no sum you could not Define your function that way do you guys get the idea you're you're saying will this converge let's find out if it does what's the interval that it will converge at what X's can I plug in that's my domain because you're defining your domain to be the ver the the values where your series converges therefore you get an output for your function if you don't get an output for your function how you can define a function well I talked a lot did that make sense too I said the same thing earlier but I want to make sure you really got it now let's talk about for general power Series so when we have a function equal to a general power Series where it's doesn't necessarily Center at zero by the way if you where is this thing centered do you understand that if I plug in C all I get is a sub Z does that make sense to you but I plug in C everything else besides the first term is gone because you have x - c if x is C if x is C then x- C would be 0 0 0 0 0 0 0 all I to have is a0 first that's defined as your Center okay so the reason why we have why we say it's centered at the origin please listen carefully is if I plug in um like to let's see well if I plug in my Center itself would I end up getting it is something right around the origin all right if I plug in the center here I'm going to get a sub 0 it's going to be centered at C comma a Sub 0 that's going to be the first point of my function now what else I want to say oh yeah the domain this is kind of important for you too the domain of your power series is always going to be some interval that is centered around C notice something what's the center here the center it's got to be a constant the center is zero the origin zero look at your domain what's right in the middle of your domain it's centered at zero that's where we get the word center from that's what Center means okay it's right the center of your interval so uh note for you the domain of f ofx the domain of your function your power series will always be an interval with xal C at the center of it hence why we call C the center now we have a few conditions for convergence of a power series I want to tell you about before we get to our example I really hope we get to our example today I'm not sure if we will but at least we'll get through this so convergence of a power series could be one of three things first thing you might have a power Series where the function is only defined at the center it could be defined just at xal C if it's defined at xal C only we would say that the radius of convergence is zero a radius has this idea of you going around something doesn't it the radius of convergence is zero it's only at one point let me make sure you understand this here real quick look at the board this has a center of zero it converges at zero it has a radius of convergence of one because it goes to 1 and one that's where Circle it kind of yeah so it's not going really up or down but it's centered at zero it has a radius of one so we go 1 and one does that make sense to you so we can have power series functions which are defined just at the center and you have no radius of convergence Therefore your R I'm going to Define r in just a bit R would be zero and the interval would be this it's a silly interval but it would be 0 is less than or equal to x - c Which is less than or equal to Z it would be the interval of converion would just be right at the center does that make sense you sure okay number B you could have an interval of convergence that incorporates or includes all xes so are there times when a power series is always convergent yes for any X yes if I had a power series with which converges for any X for anything what's the interval of what's the radius of convergence I would say infinity it says hey you know what I converges for any number therefore the radius goes positive infity negative infinity and therefore we would get an interval of convergence from negative Infinity to Infinity centered at C whever that is you guys okay with that one yeah this this one's not all that interesting just converges all the time okay this one's not all that interesting because it has no radius of convergence it converges at the center cool C's where we like I like it's these are the more interesting ones to me I like to spend most of my time here we can have convergence for values that are within a certain radius where and this is mathematically how you say this where if you're outside of that radius you get Divergence so let me uh let me make a couple notes on this one because I haven't defined this yet first thing that R is called your radius of convergence what it means is how far can you go above and below your Center remember the center is always the center of your interval of convergence okay it's always a center so says how far can you go above and below that Center of convergence and we still have a power series that works for you still has a a defined power series that's what the radius convergence is above and below now we can do this with this since since absolute value of x - c is less than R what that means is that R is less than x - c Which is less than R which means if I solve for C what we get is C minus r R and C + r that right there is your interval of convergence that tells you how far down and how far up you can go keep in mind whatever these two numbers are c will be in the middle of those because you're adding C to both sides C's going to be in the middle of these right here C's going to be right in the middle so C's going to be in the middle of this whatever your your constant is uh you can also write it this way C - R to C + r now last little comment before I I give you a uh an example on this one I want to get through at least one do you notice anything about this I said less than and I said greater than where's the equal to go greater the equal to would be the end points of your interval do you get that now I'm not say anything about the end points the end points must be checked individually so the op the way we go through these is we find first the interval of convergence we find this of course we find the radius of convergence before that that's going to lead us to our interval of convergence after that you must check whether each end point is included or not does that make sense and to do that um we use conversion tests because we've already done that so we plug in a value for x look at this listen I'm trying to give you everything without an example I know but if you had this one I said okay we have a we have a power series now plug in a value of x what that value of x does it gives you a legit series doesn't it not a power series anymore but an actual series which we can check with a convergence test so to discover whether end points are included you use a conver convergence test you might want to write that down so for end points use a conversion test for end points use a conversion test would you like to see an example I'd like to give you one before we we head out just so this is not all abstract for you yes no that CLS fast remember go for the gold Mr Leonard go for the gold that's what the really for life is okay so here we go uh you find convergence first this is going to give you the interval of convergence or Divergence or oh sorry um for Interval convergence and then you use your end points and use whatever you can to find out whether those end points are concluded included with a with a different conversion test let me give you this one so firstly what type of series power power series of course what's in there that uh would probably help us on what to do first what do you see up there factorial tells you to do something up here now listen what we're going to do is we're going to do the same test that we've just done in our last section because we still need to find out the interval of convergence we need to find out when this thing converges so what are we going to do with this problem ra we're going do the ratio test now keep in mind when we do the ratio test the ratio test says convergence absolute convergence actually when when the the limit of a ratio is less than what number shows Divergence when it's greater than one it's inconclusive when it equals to one does that make sense to you we're going to use that here so I'll I'll be upright and straightforward with you you're going to be doing the ratio test an awful lot for this stuff and the reason why is because the ratio test has a number associated with it it says that it converges when something happens does that make sense that when something happens that's going to give you your inter that you want so let's go as far as we can in the next couple minutes and see if we can do it so because we get that factorial we're going to do the ratio test listen I'm going to go quickly through the ratio test because I want to get to the punchline of this problem so ratio test ratio test would do this ratio test would put uh which a sub over so we'd have n + 1 uhhuh over what factorial very good that's exactly right so let's do it can you simplify this stuff firstly tell me something about the absolute value do you need it or do you not need it no no better better watch yourself there what's X we don't know ah is it always positive don't then you better keep it you better not just arbitrarily get rid of this thing does that make sense so be careful don't just jump to don't create your own map don't don't don't jump to conclusions here that would be bad now can we simplify this what simplifies Lots yeah lots we can do the X where's the X going to be X will be here this is x n * x x is on the numerator this do you remember do we we've done this several times you're going to get it several times more if this is n factorial this is n factorial time n + 1 so when I simplify that I get at n + 1 so what we get is a limit n approaches Infinity n Plus + 1 in parentheses * X are you okay with that one yeah strip off everything besides your X's outside of the absolute value here's why you can do it is n + one always positive yeah that's what we're going to do that's what you guys probably meant when you said absolute value you're thinking of just the ends so when we go okay well that's always positive I can do n + one outside of my absolute value and now we're going to talk about this for a second we're going to talk about when this will converge and when this will diverge so real fast I'm going to write this twice you know actually um would you remember if we got if I give this to you tomorrow uh where start cuz I want to stop right now I don't want to keep it any longer so we're working on our very first Power series and trying to show when this thing is convergent now the idea is use some sort of a test typically it's going to be the ratio test to find out where this series is converging you see the idea was and I'm not going to repeat the whole thing but the idea is this power series represents some sort of function but only where it converges so we're going to find out where this converges what our domain is basically the domain is the interval of convergence because therefore we can Define our function as the series on some domain where X makes the series converge that was all practically our last time's lesson you guys you all right with that so far so what we've done is said hey you know what that's got factorials let's use ratio test so we're going to work this down reason why we like the ratio test is because it Compares it to some number it says this thing is going to converge when the limit is less than what number let's try this again the ratio test says that this series will converge when the limit is less than what number now here's the deal please listen carefully because this is where it's coming from n's approaching Infinity right yeah so let me put in any number for X what is any what's infinity plus one by the way so this is going to be Infinity if I put in any number for x take an absolute value of it and multiply it by Infinity what's it going to be unless that number is zero I don't care if it's negative because if I take absolute value of a negative it's going to make it positive right so plug in negative a billion well then you're going to get positive a billion uh hello it's going to still be Infinity what's the one number that you can multiply by and get zero now if you're thinking well wait a minute isn't that an indeterminate form Infinity time0 not really because this is X it's not an N it's not in terms of what this is so this is legitimately zero all the time zero all the time zero does that make sense that's different so we have two situations we have a situation where if x is any number beside zero if x is any number beside zero I take an absolute value of it it's positive it's some number I multiply by infinity and what I'm going to get is infinity If X is not zero does that make sense to you yeah are you sure yes okay now if x is zero I don't care what this is how much is this going to be Z if x does equal zero now here's where the type of test we use comes into play and why it's so important are you listening yeah the ratio test says that we're going to get convergence or Divergence based on what number we get so when we get infinity all you've got to do is compare these things to one so is infinity more than one or less than one of course it's more than one so by the ratio test because Infinity is greater than one by ratio test we get uh what is it when it's greater than one we get diverence now what about this one when X does equal Z when X does equal Z we get our limit equals to zero well how's that compare to one greater than less than so by the ratio test abely we get what absolutely conver that's right absolutely convergent I don't really care so much about the absolute part but we do have convergence you see we said nothing about uh the function has to it's it's the domain is where the uh X values make it absolutely convergent it's just convergent so where it does say absolute convergence you're absolutely right because ratio test gives us absolute convergence we care more about the convergence itself does that make sense to you so by the ratio test this thing is absolutely convergent therefore it must be convergent so feel okay with that so far now here's what this says to you and we're going to do a few more examples to illustrate each type of case that you can have okay so here's what we've done we said how are we going to how in the world are we going to find the the domain for this we already know a couple things we already know a power series represents a function what we don't know is where the where is the domain that's what you're trying to find so you go okay what we find out with this is where this function is defined so where our function which represents this power series is defined all right well cool use a ratio test because the ratio test will tell you where so we work this down we got ratio test no problem pretty easy we got absolute value but n's always positive so absolute value is gone here we absolute X this is not gone because X can be negative X is any number all right well there's two situations we have if x does not equal zero doesn't equal Zer that's a number a number times infinity gives you infinity because Infinity is greater than one we say hey you know what by the ratio test that's diverent here's what that means if I plug in any listen listen carefully because this is a big deal okay if I plug in any number beside zero do I have convergence or Divergence therefore our function is not defined for any number that is not zero does that make sense to you so you go all right if I plug in X does not equal Z half a third negative 1 I don't care what it is this thing is going to diverge now the the other thing says okay what if x does equal 0 if x does equal 0 then our limit equals 0 by the ratio test that's less than one so by the ratio test that is convergent therefore for our series convergence and since our domain for our power series the function of our power series is all X's that make our series convergent we only have one value that does that what's the one value of x that makes this thing convergent that's the idea so this is one of those cases remember I gave you three cases of what you can have with your power series I gave it to you last time U what I said was that your power series could be at the Center only only at zero that's this case there is no radius of convergence the radius is zero it does there's no there's no space there it says there's only one number you can plug in to make this series conversion and one number you plug in is zero that's it there's no radius of convergence there's no interval of convergence it just happens to be zero you sure you're okay with that and it happens to be at the center itself so we get out of this we go Okay cool so by the ratio test our domain is simply xals 0 it says so right here also little side note the ratio the um sorry the uh the radius of convergence is what's the radius of convergence zero it's only one number radius of convergence is zero would you like another example that's a little bit different from this this one has only one number that you're good at it's only at zero there's no other number that works therefore our radius of convergence is zero it's nothing doesn't go above zero or below Zer it's exactly at zero show F if that made sense okay let's continue I want to do as many examples I can to give you every type of uh scenario here okay true or false this thing is a power series true or false absolutely what tells you it's a power series it's right there right there that's a power series not a problem now talking about our power series does it represent a function or can we represent this by as a function this is a function in terms of X for sure it's a function in terms of X now do we know the domain for such a function no but going to find it that's the whole idea so what we know is that this thing is a function of X when I write it out it's a function of X but it's only a function of X where it converges the where it converges is our domain so we're going to find out where this thing converges well let's let's try it um how do we do that a ratio test R let's do the ratio test ratio test is great for us for a number of reasons firstly it gives convergence and it gives Divergence right so it tells you where your series is convergent and where it's Divergent what that means is you can find your domain from that that's why we use the ratio test so much that's why I say it's so powerful so let's set up the ratio test I want you to set up the ratio test right now see if you can do it so you're going to write out what you're doing okay write out ratio test don't forget your absolute values because ratio test always has absolute values don't forget that one of these goes on top a subn plus one or a subn what goes on top okay I want to stop you right there I want to make sure that you got this let's see I I'll have you do the rest of it kind of on your own but I want to make sure you got at least this part right did you get -1 to the n + 1 here did you get x to the 2 n + 1 or the 2 n + 2 yeah you got you got n plus one in parentheses or you did 2 N plus 2 you distributed correct you didn't get 2 N plus one did you no okay good and you got the same thing here 2 * n + 1 of course we're going to distribute that no problem probably could and then factorial how many people got that one now the main denominator is pretty easy it's just this thing again that's quite a problem isn't it what's the next thing that you would do what would you do if you had this problem in the test which you're going to get something real similar to this what would you do the thetive ones very good yeah the absolute value is nice for ratio test because the negative 1es are gone so this gone this don't care what it is it's gone he gone CU negative 1 to ABS well absolute value of negative 1 no matter what it is or one it's going to give you one anyway so these things those are gone so in terms of absolute value that does not matter what else would you do here please good I'd multiply by the reciprocal changing division multiplication very good let's do that step if you haven't done already go ahead and do it now also what I'm going to do at this point you can do this too I'm going to do my distribution if you haven't done already this be a good spot to do it so here I'm going to give my X 2N + 2 over my 2 n + 2 factorial * 2N factorial over x 2 N true or false I can drop my absolute value from here to here true or false don't know FSE you don't know what oh it's squ never mind squ they are they're both squared so you can you could actually do it does it matter no not really I'm going to put them just just to get us in the habit okay so that we don't automatically drop them I'm going to put them but yeah you could if you see this you could factor out that square do you see it you could do it you just fine I want to put them just in case so if you can make it that far you feel okay with it now what would you do simpli simplif simplify as much as you can now uh when we simplify this thing this is something you're going to have to be pretty good at being able to simplify x 2 n x 2 N plus 2 you got to know what that plus two actually means here okay you got to be good at simplifying 2 N factorial and 2 n + 2 factorial you got to know what this means so I'm going to go through it one time with you from here on not I'm probably just going to kind of simplify on my own because I'm going to expect that you understand this uh let's consider this first this is the easier of the two I'm going to write up here and I'm going to erase it so if you're going to write it down write it down quickly X 2N + 2 I can even write it down jeez x 2 n + 2 over X 2N if you know what this is this is X 2N * x^2 x 2 N you multiply common bases you add powers therefore I can split up any addition of powers by multiplication comma bases that therefore gives you X2 feel okay with that one so when I do this I'm going to go okay well boom boom X2 so far so good now this one's a little more tricky to think of it's not hard it's a little more tricky to think of 2 N factorial over 2 n + 2 fact faial but think about what 2 n + 2 factorial would be do you understand that when you're doing factorials you take the first term and then you multiply by the term right below it so basically you multiply by the number one less than it so if I had five factorial it's 5 * 5 - 1 * 5 - 2 * 5 - 3 so this would be 2 n + 2 that would be the first term does that make sense yeah then I would have 2 n + aha does that make sense then I would have 2 N and then I would have oh but wait a second this is 2N factorial all the rest of that stuff is 2N factorial so fans you understand that concept we've used it before but now we've never had a plus two so let me assure you you can do this with a plus anything it just is 2 n + 2 * 2 n + 1 * 2 n * 2 N - 1 * 2 but that's that's this that's exactly what this is so basically this gives you 1 over 2 n + 2 * what times what 2 and that's it are you with me on that one that one's kind of tricky for people I want to make sure that you're okay with that yeah are you sure so when you simplify something like this you okay what's what's here what's here well this is this is 2 N Factor this thing right here is 2 N factorial with a little extra with two extra terms the 2 n+ 1 and the 2 N plus 2 and that's it for sure you okay with that one BR up say what now you just split it up you split it up you go well what is what is this thing really it's really 2 n + 2 * 2 n + 1 * 2N oh but this the rest of this is 2 N factorial so I can cross out all of this with this and then I get 1 over 2 n + 2 * 2 + 1 and that's what we're going to do here now I got to move back over [Music] here so let's see what we have left over can you tell me CU I'm going to get lost over there so what what do we got X2 over uhhuh abute okay absolute value sure do I need the absolute value anymore for sure I don't that's an X squ there's no way it's going to be negative uh this is both positive there's no way that's going to be negative so I can drop my absolute value right now if you want to now here's the deal here's a deal at this point at this point we're going to see what happens when we let our n Go to Infinity we've already simplified everything we can right if we go let our n let our n Go to Infinity what are we going to get now remember something about this you're going to give me a number for X right yeah so pick any number I don't care what it is but it's a finite number does that make sense it's anything take any number and square it do you understand that's still a finite number now divide it by what this is going to what's this whole thing going to infity Infinity so what's this divided by this for no matter what x is right oh yeah equals zero do you understand why it's four no matter what x is cu X is not changing as n is changing n is going to Infinity X is not X is not b based on n x is X is this x say uh pick a number but it's it's a it's one number it's not going to be a variety of numbers it is called a variable I understand that but you pick one number you say x is now let's see about 3/4 okay X is now 1 million okay X is now5 okay but no matter what it is you're going to stick with that number you got it then you square it you stick with that number then you divide it by something that's growing to Infinity if you div divide it by something that's growing to Infinity you have a finite number over infinity and that idea is zero and this works for all X here's my question if this works for all x what that says is that are you with me yes say I don't care what you plug in for X this this series will converge does that make sense to you yeah now how do I know that this thing's going to converge how do I know that what test did we use and what answer did we get with our ratio test and that is less than one perfect so we say this is less than one so therefore it converges by the ratio test and what we've made our determination is that this is for all X now if you have something that converges for all x what that means is that I don't care what you plug into right here this this power series will converge for anything we've just proved it does that make sense to you cu no matter what you plug in hey guess what it's going to be zero cool well that means that we converge for all X if we converge for all X our radius of convergence says infinity infinity so the radius R would be infinity and our interval of convergence would be well this is hopefully pretty common sense says if you can plug in anything right your radius is infinite if you can plug in anything our domain is negative Infinity to Infinity remember that the domain is all X's that make our series converge if we can plug in anything to make our power series converge then our domain is infinite negative Infinity to Infinity so if fans feel okay with that okay we've covered two out of three three scenarios we've covered one where we just converge at one number just at the center and that's it the radius would be zero we've covered the next one where we converge at any number I don't care what it is our radius would be infinite and we'd have netive Infinity to Infinity let's talk about the third example question um if you can plug in Infinity into X but you can't okay so it's everything except for actual well X you're Define X to be any number right but you can't say infinity for X because that's not a number we say plug in anything negative infity to Infinity of course you can't get to here that's why we use our parenthesis that's why in our interval notation you don't use a bracket because the idea of going to Infinity is a limit right kind of accomplish that X does not get to go to Infinity in this case n is going to Infinity in this this case does that make sense okay that's a good question any other questions before we do our our next example we're going to do several more but I want to make sure you're good and N is different or Infinity is different for n because the idea of our limit here is saying that our n is growing our terms are going to Infinity our X doesn't change though right so once you start this you pick X you stick with it so it's first term then first terms times an X then second term * an x to the second third term * x to the third fourth term * x to the 5th but the X doesn't change the x is one number so if we say I don't care what number you pick stick with it the series is going to converge that's what's going to happen here no matter what x is no matter what x is that power series converges therefore this represents our function for all X you still are you sure you're with me you to do one more yeah so the first example we had today the one we we just ended that represented a function represents that for only one value of x that value of x would be zero a function represents this power series for any value of x all the time it's going to converge for anything let's look at our third and final example third and final scenario iose a seemingly harmless example these are these are a little more fun though these are the ones that I like okay the first one well it's it's always going to equal well the only time it equals z is if x is zero okay this one equals zero all the time it's less than one all the time for any value of x that that's kind of boring let's look at this one look at what happens here so uh first thing what type of series is that uh Power series that's right it's a power Series so okay if it's a power series uh well we need to find out where the series is going to be convergent in the first example it was only at zero in the second example it was anywhere in this example we got to discover where this thing is convergent because only where our series converges is where we can define a function for it only on the domain so first example domain was x equals 0 second example domain is infinity infinity to Infinity third example we're going to discover where that is it's going Tak us a little bit of time on this one so what would you do first root test you could maybe do the root test if you wanted to I'm going to encourage you to stick with the ratio test uh this is going to be a little hard to do the root test because you don't have a power of n there so it'll be a little little difficult so i' probably stick with the ratio test still ratio test works for powers of n as well so try out the ratio test go ahead and set up the ratio test if you would question you don't want to use the limit test we don't know what x is no the ratio test right there will give you where this converges and diverges which is so why it's so nice right because it says well it tells you whether converges or diverges therefore it will give you the interval here of convergence and Divergence so we can't do really a like the Divergence test you wouldn't use that here you use a ratio test to tell you where it's going to be convergent and where it's going to be Divergent okay now after after we do that I'll show you when you can use something like a diverence test okay so we're going to use the ratio test and I'm hoping you're already working on it ratio test says that hey do we need absolute value for a ratio test yeah because that n's changing this is going to be possibly if you give me a Negative X positive and negative you need the absolute value on the numerator you need to have x n + 1 over n+ 1 that's right and the denominator you need to have x to the N over that's it n your absolute value this one's going to be kind of nice we have a limit we have n approaches Infinity are we still going to have our absolute values yes absolutely yeah that's right and we got x n + 1 over n + 1 times n x did I get that right yes okay what would you do now simplify as much as you can also also collect your n terms and collect your X term you'll simplify the X terms okay okay there's going to be only one X left collect your end terms pull those outside of your absolute value because they're always positive anyway does that make sense to you so what I mean by this is well simplify what you can but don't do stupid things don't go oh well n and n cross out I'm done don't do that don't do that simplify this sure x to the n x to the n + 1 gives you one extra X that's what the plus one is it's x to the first Power this you don't simplify right now this you're going to do a limit with in just a minute what I mean is collect this collect it and say okay well that means that we got a limit as n approaches Infinity of n/ n + 1 * X in absolute value are you still okay with that yeah you sure yeah now is n / n +1 always positive yes because n's growing from 1 to Infinity yes it is is X always positive we can't make that jump I don't know what x is so what I can do this is what I've been telling you collect your n terms pull your n terms out of your absolute value if they're all positive they generally will be so we're going to have n/ n + 1 time the absolute value of x now this is why I like these it's a lot more interesting you got to do a little bit more work with it is it super hard no it's not super hard but you get to do a little more work with it so please pay attention this is how you do the rest the first two are kind of our basic ones okay X is zero only for zero no radius X is infinity radius INF X is I'm sorry X is zero for all X therefore radius is infinity no problem this is more interesting take the limit as n approaches Infinity of n/ n Plus one what you get how much you get do low I don't care what you do divide by largest power of n whatever you're going to get one this whole thing is one does that make sense to you what is X as n approaches Infinity it's absolute value of x it's whatever you have here so what we end up getting listen please we get 1 * the absolute value of x we get the absolute value of x okay I need to show up hands if that makes sense to you notice that the the X doesn't change it's the N that's going to Infinity therefore we have this idea of a limit of n like we normally have we just have tacked on this absolute value of x now here's the deal this is why we use the ratio test you see the ratio test tells you when you're going to be convergent this is the whole deal so if you get this you're going to get the whole thing okay the ratio test tells you when you're be convergent when does the ratio test tell you that your power series is going to be convergent when the limit is less than what when what's less than one absolute value this not X the absolute value of x so did did you guys catch that we're using the ratio test right yeah ratio test says hey go through all this crap and at the very end if this thing is less than one then you will have a convergent series or you'll have a convergent power Series in this this case the reason why this is here why the less than one is here is because we just use the ratio test show fans if that made sense to you that's why we're doing it folks now this is kind of cool that's just very cool give me a specific value of x that you could plug in that you you know that this series is going to converge forward give me a specific value2 would be great zero would be fantastic what else ne one2 how about two no that's bigger than one how about one itself how about negative one itself no yes I don't know I don't know the ratio test does not tell you so here's what we know so far did you guys catch the whole idea about if you have a value between 1 and one between not inclusive but in between you will for sure have a convergent power series then here's what we know already because of this case because absolute value X has to be less than one by the ratio test we know that -1 is less than x Which is less than one we know X basically has to be between 1 and 1 now let me tell you something right here is where you find your radius of convergence it's really easy to do please listen carefully what you need to find your radius of convergence is what whatever is being raised to the nth power x in this case if it was x -1 to the N you'd look for x - one that's look really nasty I know but if you had x -1 in there you look for x -1 here and whatever that number is that is what your radius of convergence is so in our case our radius of convergence is what number is this what's a radius of convergence there you go so you look for whatever's being raised to this power right here when you get that that is your radius of convergence so the radius of convergence is one so so far what we know is that yeah our power series represents a function no problem we've used our ratio test to say that this thing this power series is going to converge when the absolute value of x is less than for sure so basically when X is between1 and 1 what that says is the radius by the way um what's the center what's the center of this zero origin zero the or okay zero Hey add one to zero what do you get subtract one from zero what do you get that's why the center is zero the radius is one okay if you and the way you do this is you find out the thing that's being raised to n that's your radius that's what it is now the issue is are the end points included or not well the ratio test does not tell you that you have to determine that so now for our end points it's not hard in fact it's all that stuff that you learn about series which is kind of cool so now for end points um can you tell me what are our two end points that we have here and one got to check both of them there's only two things you got to check two two things you got to check there's only two end points so we're going to check 1 and we're going to check one go wait a minute how in the world can I check my end points well let plug it in I just spit all over do you see that gross plug it in here so here we go so if we have our series n equals I say one should that one is one interesting so if we have our our Series going from one to Infinity of x to the N Over N uh what am I plugging in for am I plugging in for the N or am I plugging in for the X X it says X so what we get is a series from one to Infinity of oh it can't be zero we divide oh I'm sorry what what's our X going to be wait a minute look at that okay so if we plug in negative one what's it do it changes our power series into just the typical series that you have seen before this will change into something you've seen before tell me about that one Al alternating harmonic we've done alternating harmonic before it converges so this is an alternating harmonic series and because we've done this already before we know this thing converges if you listen carefully if you didn't know it converges you would do one of the tests you've already done you would do alternating you wouldn't do absolute convergence here because if you do absolute convergence you get one over n and that would be Divergent but remember that when you're doing absolute listen carefully to this one okay when you're doing absolute conver if you take the absolute value and the series you get diverges it doesn't mean the whole original series diverges it means that it just doesn't absolutely converge remember that idea about conditionally converging was the first thing that we talked about okay so here you go well let's do the alternating series test the limit is zero it's clearly decreasing alternating series test says converges so even if you didn't remember this we get it convergence now the next one I'm going to move it up here didn't take a lot of work the next one if we get a series Nal 1 to Infinity of X the N / n and we say let's let our xal 1 then we get the series from n = 1 to Infinity of 1 to the N Over N what's thaton 1 to the N is always one isn't it what's that again tell me some about harmonic so we only [Music] includ that my friends is how you find out whether or not your end points are included so here's what you do for the most part on all these problems with your power series it's not that hard I know it looked originally like oh my gosh what are we doing here it's not that hard honestly what we what you're doing you're taking power series you're going okay I know that represents a function where use a ratio test the ratio test will tell you where remember ratio test only converges for whatever you get here is less than one so ratio test is cool man you know where that's going to converge you just don't know the end points use those end points in your original Power series with some of those tests you already know alternate hon converges harmonic diverges what end point will we include what end point will we not include one we include good and one we don't include one okay so now the last thing we're going to do here so last thing we're going to do so our radius what was our radius of convergence and our interval of convergence is what's our interval of convergence bracket why is there a bracket there1 makes that series converg remember our domain is all the numbers that make our series converge so if it converges negative 1's included if one diverges one is not included we already had the rest of it right here so we go negative 1 to one and just talking about the end points now does that make sense by the way what was the center for this zero where's the center between this does it make sense tell you what let's go ahead and start one more this is practically we have two more examples and that's about it then I'll teach you how to do calculus of power series how to do derivatives and integrals of this stuff yes you can yes you can you can do it don't fret don't fuss it's calculus do it so dorky I can't help it oops that can't be right there we go oh dear all right let's do it let's go through it right now uh firstly what what is this you see something like that with X ra to n that's power series absolutely power series for sure now you tell me right now where is this power series centered at two very good centered at two which means this when we get down to this part our interval of convergence the center is going to be at two it'll be two if it if it converges it's going to be at two if it converges a more than one number it'll be two plus and minus the radius of whether that's infinity or that's a finite number does that make sense to you but two will be at the center for sure okie dokie what are you going to do ratio test do ratio test that's not power ratio test is going to give us something less than one anyway do a ratio test what are you going to use for most of these things ratio do you understand the ratio test works when the root test works as well yes do the ratio test so we're going to State what we're using and then we use it be real careful on getting these correct here be real careful of course the a sub n is pretty easy because it gives it to you but don't make a mistake on the the n+ one idea uh remember the a subn always goes on our main denominator here so we're going to have x - 2 n/ n² * 3 n you forget one little thing and this problem explodes in your face okay we're going to get that in a second okay so up here we have by the way when you doing the N plus one does The X ever change so this guy is going to beat there no matter what x - 2 the N does though we get n + 1 over n + 1 2 * 3 n + 1 oh goodness and I'm forgetting something what else am I forgetting yeah you're right is that important yes an absolute value forget your absolute value look at what happens look it if you forget absolute value you lose this don't you you lose your other side of your interval that's a problem okay so oh my gosh from here is there anything that we can get rid of automatically right now no there's no netive 1 is not like that alternating idea so we'd have to go through and multiply by the reciprocal as and yes absolutely you got to simplify as much as you can do we drop the absolute values right now no oh no the x is for sure we can't drop that so we got x - 2 to the n + 1 we got n + 1 2ar we got 3 n + 1 no problem now multiply by the reciprocal we got n^2 * 3 n we got x - 2 to the n and then we still have absolute value I I promise you the simplification gets way easier when you get used to this right off the bat you're like oh my gosh what are you doing but when you get used to it it gets really quick it'ss really nice you get used to what a plus one and a plus two mean as a power and you can simplifly stuff really fast have you already done it yeah probably tell me something that does simplify yeah that's the first thing I see two so if I simplify 3 to the N with 3 n+ 1 I get and it's on the denominator yeah yes how about the n s of the n+1 squ no you can't touch that can't touch this okay don't do that I'm not wearing my hammer pants but I should be um you can cross out the xus 2 N the Bott sure sure these are the same except that this has an extra x - 2 so if I do this and have the extra x - 2 then what we end up getting you know well we're not going to have much time after in this anyway so I'll race this what we end up getting is a limit and approaches Infinity do I still need the absolute value what's on my numerator please [Music] over okay x - 2 no problem n s no problem 3 no problem n + 1 squ no problem show hands feel okay with that here's what you do now please be careful what you do now you collect your n terms you do not collect constants and X's those are important so collect your end terms what I mean is that this stuff I'm going to move this off to the side I'm going to get a limit see if you can stick with me here n/ n +1 squared do you see it n / n +1 squar notice how I don't have an absolute value it's always going be positive then I have an absolute value of x - 2 over 3 collect everything except for your X values and your constants does that make sense to you yeah honestly are you guys okay with that how much is n n + 1 as n goes and what's one squ one so this whole thing is no so what is this whole Li actually I'm going to do this for you does this absolute value affect the three at all no that's what we get show should be okay with that so far now here's the question what test did we just use raest when does the ratio test show convergence when the limit is less than again this is by ratio test that's why we're doing it because ratio test says that our power series any series converges at the limit is less than one no problem so this power series make the limit less than one therefore you guarantee this thing is going to be converted does that make sense definitely have that okay now what that tells us is that if I multiply by three If I multiply by 3 what I end up getting is absolute value of x - 2 is less than 3 you guys okay with that one if you're wonder why are you multiplying by three now here's what I'm trying to do a lot of students get confused how in the world do I my radius of convergence is my radius of convergence one is it three is it five here's how you tell see the thing that's being ra to the N power over there the x - 2 as soon as you isolate that as soon as you isolate that x - 2 x - 2 in absolute value as soon as you isolate that that is your radius does that make sense so the radius is perfect radius of convergence is three that's what this says right now so I'm going to put this right here we already know the radius of convergence is three now from here all you got to do is find the interval of convergence and it's so easy it's really easy all you got to do is solve for x so if you solve for x what an absolute value inequality means is you do -3 is less than x - 2 is less than 3 that's what an absolute value inequality is right there then if we add two we find our domain3 if you simply add two so be okay with with that so far so this is why you do your radius first because right now it's kind of hard to see it's not super difficult you can still do it by the way what number is at the very center of this two 2 + 3 five 2 - 3 Bo yeah radius is three Center is two now what about the end points can we do that real quick we do it I'll do it here um you know what I'm going to race this I'm going to do it over here we're going to test your end points here so last little bit end points so for our end points we need what's what's our first endpoint we checked don't be silly and check this that's silly okay make sure you solve for x to find your domain this is the end points of your interval right here so we do1 and we do five so if I'm plugging inga1 what we end up getting remember I'm plugging in negative 1 right here we end up getting -3 n over n^ 2 * 3 to the N what that gives you if you look at this real carefully what this gives you is what kind of it's kind almost you're so close you said one thing wrong but almost do you notice how they both have the power yeah so you can make -3 1 over 3 to the N right geometric Time 1 n^ 2 no not geometric do you see that this is3 3 to the n * 1 n okay it's on denominator how much is33 -1 to the n * 1 n 2 N = one to Infinity tell me something about this series Ser it's not a p series Al alternating series you can do a few tests with this one if you wanted to if you want to be really quick about it if you want to be real quick about it do absolute convergence absolute convergence gives you 1/ n s yeah P series booah done this series converges absolutely if you don't need absolute convergence which you don't you just do alternating series test so that's what I'll do because I don't need absolute convergence I'll do by alternating series test it's pretty clear that the limit is zero it's very clear that this is decreasing limit zero decreasing alternating series by alternating series test convergent are you guys okay with that one could you have shown it a different way absolutely you could used absolute convergence done absolute value this is gone 1 n s is a p series because the absolute value of series converges our series is absolutely convergent therefore it is convergent end point no matter what is included number uh plug in five if we do that we get n = 1 to Infinity of 3 to the N over n^2 * this one's even easier how much is 3 to the N over 3 the n one one one over n tell me something about 1 over n s say what now that's a p Series yeah tell me about the P series P oh get too fast P series P equal 2 that's greater than one do you notice how what we're doing look it we're putting like everything we know together right I'm going to go through it one time here real fast to show you what we're doing we're doing ratio test to give us an interval that lets us know when our power series converges when it represents a function we're using all the stuff that we've learned already to do our end points we're saying hey alary series test or absolute convergence or a p series or a geometric or whatever you have to show that our end points are or are not included so here's what we got the radius of convergence is three the interval of convergence is -1 to 5 we already have that now tell me about the end points both brackets that's what we're talking about here's what it means it means that this series represents a function of X that converges for 1 to 5 and that's it make sense that's F all right so let's do it the last example before we get on to the calculus of power series uh from before we know that that right there whenever we have an X raised to a power of n we have this thing that creates a function of X that's our power series what we are focused on doing is how to find the interval of convergence the rat convergence where this thing converges firstly notice something about this power series where's the center the center is zero sorry Z zero that's right which means that whatever interval we get it's going to be centered around zero plus or minus some number whether that radius that number is zero infinity or an actual finite number this interval of convergence is going to be centered right at zero and you found three cases so far you found those cases on your homework where it converges only when x equals 0 and diverges elsewhere that's the most limited most limited amount does that make sense you found the cases where it converges at the center and some radius around it where that radius is a finite number and then we have to check end points of course where whether it converges to end points or not that used the stuff we learned before and then that's kind of the the second case the second least limited and then the least limited is where it converges all the time for any X and you hopefully if you've done your homework or some of it you've seen all three of those cases have you seen those cases yet by the way I do want to see where these things converge they will converge somewhere okay I want to see that I want you even if they go it diverges when X doesn't equal zero I want to see it converges when X does equal the center wherever that happens to be does that make sense okay so our last example says all right if that's a power series let's go ahead let's find out what the interval of convergence is what the radius convergence is what's going on with this thing how are we going to do that most of the time with this yeah usually it's a ratio test or maybe a root test but you want some test that lets you determine the interval or the radius convergence that says hey it's going to converge when this is less than some number when this happens because that gives us that interval so let's go ahead let's start do you guys want to try this one on your own see if you can do it let's start with the ratio test write out the ratio test and see if you can do it as you're working through this can you tell me some of the key points about the ratio test that we need to know say what no it would take away the alter it would why what what part of the ratio test takes away the negative one absolute okay so ratio test has absolute value in it no problem uh what are some other key points about the ratio test that we need to know specifically what goes on the main numerator a subn or a subn plus 1 okay that's pretty good do you know anything about the convergence of the ratio test when does the ratio test converge very good when limit is less than one and it diverges when the limit is greater than one this is going to be a nasty one isn't it let's see if you all got this we got1 the n + 1 we got 2 to the n + 1 we got x to the n + 1 we got Square < TK of n + two do you see where that's all coming from yeah all right cool we're just taking n plus1 and put in here and here and here and here and then all over a subn so-1 the N not a problem 2 the n x the N Square < TK of n + 2 show hands if you made it at least that far I'm sorry one hopefully you didn't make it that far I just like twos today n plus one did you make it that far yeah fantastic you know the ratio test all right so now what do we do as much as you can so simplif means get rid of your1 n + 1's or 1 n because absolute value gets rid of those things it also means take your complex fraction and rewrite it do I get rid of my absolute value with a ratio test in a power series no not generally unless you have something like X squ where it doesn't matter sure but not generally no so we're going to have absolute value still we'll have 2 n + 1 * x n + 1 all over < TK of n + 2 times the reciprocal of our main denominator that would be the square root of n + 1 over well the negative 1es those are gone our absolute value does that for us we get that did you get that far on your own yeah okay now before you go any further what are we going to do yeah as much as you can simplify and then after we simplify we're going to strip off our n terms and leave our X terms we're going to separate those things so when we do this I'm going to go through it quickly I've always explained it a few times to you when we simplify you're going to have a lot of common bases raised to different powers just understand that a plus one simply means you're multiplying by two a plus one here means you're multiplying by an X so what we're going to do we're going to collect our n terms if you notice this here's a < TK of n +1 over squ < TK of n + 2 I can make a big square root of n +1 n + 2 we're going to do that so n + 1 n + 2 as the radican of our square root and then we got what else is up there abute value 2 2x okay if this is all an absolute value tell me what do I do what can I pull out of the absolute value right now because it'll never be negative good how about the two could I pull that out you know what could I pull out the two I could pull out the two so if I wanted to I could do a limit as n approaches Infinity of the sare < TK of n + 1/ n + 2 time 2 time theun of X I could do that absolute what did I say oh man sorry too much painting I told you brain kind of my bad so square root of n plus I just like square roots in Tuesday don't worry about it squ of n plus 1 over n plus 2 * 2times the absolute value of x oh my gosh tell me one more thing that we got here what's going on Li the limit what's a what's the square root of n + 1/ n + 2 as n approaches Infinity inside of our limit what's this do yeah use all that stuff about limits that you know n over well if you divide by largest power of n or you do low TS inside your square root whatever you want to consider this whole thing goes to one so what's our limit equal very good okay let's put the whole thing together now what test did we just use Rao and when do we know that the ratio test converges with the limit is less than one so if we just use a limit and say hey you know what we use ra chest to find our limit and we know that when the limit is less than one we know that this power series converges remember that a power series represents a function only when that power series converges so that's the whole idea here behind you is the ratio test is we want to find out when that actually converges so we know the domain of our function the domain of our function for which our power series converges show F feel okay with that idea I know explained this a couple times I'm make sure you're with it do I stop here no no no but we're we're getting there what we know is that 2 absolute value of x is less than one gives us our interval of convergence for our power series for which our function will Define that that's our domain now what we do here is we always trying to find the absolute value of whatever is being raised to the N power isolated all by itself on one side of our inequality what that does that will give you your radius of convergence have I told you that before yes so what you're looking for here is hey if it's x to the N I want X all by itself inside of an absolute value on one side of my inequality so if I divide by two on both sides that right there as soon as you isolate they go oh cool look at that hey this is what's being rais my power it's isolated it's my inside my absolute value right now you should be able to tell me the radius of convergence what is it very good so this right here says the radius of convergence is2 now we keep going a little bit we we write out the definition of what an absolute value inequality is absolute value inequality says if you have the absolute value of x less than 1 12 that means that -2 is less than x is less than 12 and we're almost done are you guys okay with that one yeah almost done what do we do next check end points we got to check those end points because right now yeah we have the radius we know it's going to be 1/2 we have the center notice how the center zero is right in between 1/2 to 1/2 what we don't know is whether these two end points are included or not that doesn't give us this is the ratio test it doesn't give us it by just doing the simple mathematics we got to check the end points individually where do we check the end points say what origal original one so plug in in let's see if we plug in where we where do we plug in the 1/2 and thetive one2 it seems that way right because X is between those things so if x becomes one2 we're going to find out whether that converges or diverges for that specific end point same thing with positive one2 which one do you guys want to do first Okay negative does it matter no no go left to right so if we're going to check the end point - one2 we're going to plug it in here and get a series N = 0 to Infinity of1 to the N we're going to have 2 to the n and we're what's what's the next thing yep all over the square root of n+1 now because you did all that stuff in the previous sections these on should be pretty easy these are going to be nice things that you recognize oh yeah we could do this with it we can do alternating series test or absolute convergence or we could do another ratio test if you really really want to or we could do a p series or a limit comparison test all those things open up right now because we don't have any more X's we have all constants so simplify before you check them but it's going to be something familiar so here's here how I would look at this one nth power nth power nth power you see if I can make all those things to One n power yes well then we have 1 * 2 * -2 how much is1 * 2 * 12 POS 1 so this whole thing gives us a series from 0 to Infinity of positive one to the N over the < TK n + 1 okay well wait a minute how much is 1 to the n one so no matter what do you guys notice that when I have a negative time a negative that that gets rid of that idea of alternating series for this is not alternating it's just one to the N or simply one all the time so we don't need that n anymore now tell me something about this series can you work with it can you do something with it compar okay if we compared it please listen carefully I want to walk you through this because I don't want you wasting your time you need to think about it and and I like that you're thinking about it but if you compared it this is less than 1 n correct and this is 1 n 1/2 which is a Divergent P series are you clear on that yes this thing diverges this is less than that does it tell you anything this is inconclusive that does not help us at all so even though we can compare it we're comparing it to something that is bigger and Divergent that doesn't make sense we want bigger and convergent that would be great but this is not that does that make sense could you use a limit comparison test that you could do if you did a limit comparison test then the series of 1 the n + one compares to the series of 1 overun n sure no problem if you did the limit since these both have positive terms we'd have < TK of n + 1 over 1 n as n approaches Infinity this should kind of be old hat for you right you guys have seen this before I know and this would give you one more I mean when you work through it that's going to give you one quick Head n if you're okay with with that one now next question if that gives you one this is a limit comparison test is that inconclusive or do you know something about this you know something you know something with a limit comparison test if the limit equals a number I don't care what it is if it just exists if the limit exists then both both of these Series have the same result tell me something about this this series tell me something about this series then so by the very good by the limit comparison test we said this all right this one acts just like that think of the end behavior of your function this one acts just like this one if I compare them in a limit a limit comparison test them in limit and that limit exists they both have the same result if one diverges the other diverges if one converges the other converges so we base this our given series on a series we already know the result for whether it's convergent or Divergent doesn't matter they're going to have the same result so both series diverge by a limit comparison test since that was a p series with P = 12 oops okay so let's put this into into play here is the end point negative one2 going to be included in our domain of our power series okay so since this thing diverges we are not going to include one2 can I show you something else a different way to think about this it's a little weird but I want to open your mind to something a little bit different you don't actually have to use a compar test to do this I'll show you this one check it sometimes you can manipulate your Series in such a way to change it from what it is to what you want it to be and here's how you would do that in this case so notice something if we start with n equals 0 what we're going to get is a square Ro T of 0 + 1 do you get that well you can manipulate these series as long as you start with the same thing and it follows the same pattern for instance instead of doing a series from n 0 to one Sor sorry 0 to Infinity if I start it at one then what changes about this is instead of have n + one inside of my square root all I need is something that takes this number and Maps it to the same place that this did so for instance here's one here's one here's square root square root here's n here's n here's plus one okay when I plug in my zero I get zero right when I plug in my one I need to get zero n minus one does that make sense to you it's doing the same thing think about that if I plug in zero here I get uh one over 1 if I plug in one here I get 1 over 1 if I plug in two here I get 1 over 3 plug in two here I get one over three sorry uh what did I say two here yeah right so plug in the next one it maps to the same fraction well if that's true then this becomes tell me something about that one so this becomes the same exact Series this is a Divergent P series you can do it a different way I just want to show you that one is it super important probably not you can do a limit comparison test and get the same result but I wanted to open your mind just a little something else so hands if that made sense to you okay cool we're almost done what's the last thing that we got to do here yeah let's do that one let's do the positive one2 so if we do2 the series will look really similar except instead a negative 1/2 we get positive2 what's going to happen with our numerator let's simplify that what do we get a that's right we we don't have that negative we had here so when I had negative * negative gives it a positive I don't have that here so our series does look really really similar it's still going to be a one but it's it's a negative one to the N over the < TK of n +1 now this one oh man this should be ringing some Bells tell me something about this series that you know right now it's alternating all right so alternating series we take the limit what's the limit of that series Zer Z remember doing that limit is n approach Infinity is zero if the if the sequence from which we get our alternative series goes to zero that's like the Divergence test for our alternative series next up is our series well what's the next thing have to be is it decreasing term by term yeah and that's really easy to show so we'd say the limit of 1/ s n + one that's our a subn so to get our a subn you cover up this you go okay what's remaining after I take away the alternating part that's our a subn if the limit as n approaches Infinity gives you a zero that's like passing the Divergence test and saying okay it's not maybe not Divergent you keep on going after that you say is it decreasing is a sub n +1 which is 1 < TK n + 2 is that less than 1un n + 1 which is our a subn if that's true which it is here this is definitely less than that for n greater than equal to one this says it's decreasing so by the alternating series test we get convergence let's put the whole thing together then uh do you know what your interval of convergence is oh interval your interval of convergence do we know what it is right now yes where's it start ladies and gentlemen is ative 1/2 included no no where does it end is a 1/2 included yes so we know the radius is 1 12 we know the center is zero we know our interval we know everything about this so for our last example I want to make it real clear make sure you guys are really good at this we got these things called Power series which are functions power series have an X in there so when I write out the series I get a function of X now that function is only defined where that series converges that definition that that's our domain so what can X be in order to make this thing convergent that's how we go through all this mess we go okay well what what's X got to be for that to work what's X got to be for a series to converge so that Define a function of X we get all the way down to here we say okay whenever I have my whatever's being raised to the N power isolated inside my absolute value this right here is your radius if this had been an x - 3 I would want an xus 3 here before I determine my radius does that make sense to you after that after we determine the radius that's all fine and good we get our interval and then we check our end points the end points are going to be done with something you've done before in this class uh either an alternating series test or a limit comparison test or another test that we have integral testing I don't care what you use as long as you can test this thing out if you get for an endpoint that a series diverges that endpoint's not included in your domain that makes this power series convergent so defines our power series if you get that the series when you plug in that n point for X does converge then the end point is included and this is what we're looking for the radius the center and our interval of convergence that makes this power series work makes it converge therefore we can define a function by by using that domain have I explained everything well enough for you guys to understand it are you sure I don't want to leave anything hanging do you guys get the idea can you do it yourselves okay then the last thing we talk about in this section is how to actually do calculus with these power series which is weird like what uh calculus what do you mean how do we do derivatives and how do we do integrals of power series and honestly it's not that hard but it leads us to something very very interesting it's kind of like our our stepping stone between this section and the next section which is Taylor polinomial and mclen uh polinomial well well Taylor series and the chor series that leads to [Music] Taylor but firstly we got to learn how to do derivatives and integrals of power series well how in the world are we going to do that well you know what let's start with something general let's just just build on it so let's assume that we have this power Series so f ofx is a power series starting at zero going to Infinity a subn xus c to the N let's talk about the terms of what the series is going to look like so we're going to say here that we have some power series it will converge for some domain of X therefore we're going to have this function so where our power series converges that's our domain for our function okie dokie now what do the terms look like what's the first term yeah a sub what a a sub Z how about xus C is that going to be up there no no because it's being raised to the zero power that gives us one so we just get a sub Z Plus what what the next term going to look like sub one very good to the plus what uhhuh power good now we're getting the pattern a sub 3 x - c to the 3r and it doesn't end well here's the cool thing um you know about derivatives right derivatives say okay well take your take your function and when you do a derivative we basically just derive it term by term don't we and when we have pluses or minuses we know that with derivatives we can separate those pluses or minuses and just do the deriver of that term put it back together with the sign you're good to go well if you think about it when I do all my terms tell me what a sub Z and a sub 1 and a sub 2 and a sub 3 and a sub 4 and all my a sub ns are in reality are they variables or are they constants so no matter what that's going to be a constant that's going to be a constant that's going to be a constant that's going to be a constant and so forth and so on the only place I have things to derive because I have a function of x if I'm trying to find frime of X the variable is X here the only place I have my variable is here and here and here and so forth and so on does that make sense to you so we notice that when we take derivatives here am I going to need a power or sorry product rule here no that's a that's a like a coefficient okay it's like three or seven I don't care what it is it's a coefficient so what we do with this is yeah if we have just a constant by itself what's the derivative of a constant okay so we're going to take a derivative term by term the derivative of a sub Z is gone I don't care what it is it's a constant when der that it's gone so frime of X gives us firstly zero I'm going to write the zero just so we see what what's going on with that okay say derivative of a constant Man Zero what's the derivative of this guy what's that going to be not one no if this was 3x - c it wouldn't be zero it would be 3 * 1 x - c to 0 power * the derivative of the inside which is one and this is one this is please don't forget your basic calculus okay it's just the chain rule folks it's you bring down the power you multiply by the inside good to go so this would be in this case what don't say three it was just an example a sub one that's right a sub [Laughter] one what Okay Plus what are you going to do with this if that's a coefficient like three you're going to bring down the two aren't you you're going to have two * a sub 2 you're going to have your inside x - c you're going to raise it to 1 power less than what you had to the first Power and you're going to multiply by the derivative of which is one all the time it's centered at C it's just going to be xus C it's one all the time so that's great so we're using using the chain rule yeah but what we're really doing is saying okay well derivative of a constant is zero that's a coefficient I know that some you go well wait a minute a constant der of constant zero well yeah it is if it's by itself with no variable if not then you're using kind of technically using the product rule all the time it's just the derivative of a constant gives you zero so you emit half of it and you're using the chain rule you're bringing down your power you're multiplying you're taking one away from that and using the chain rule multiply by der inside the Der inside is always one you with me yeah so tell me what the next one's going to 3 and so on and so on and so on and so on forever and ever and ever but look at what this really this is so cool the way this actually works check out what happens so F Prime of X really just does this um first thing tell me where our series actually starts now you see our series here started at zero we had a Sub 0 where's our new our derivative start a sub 1 it doesn't start at a Sub 0 anymore it starts at a sub 1 why well because the derivative of our very first term gentlemen uh gets rid of that constant so this thing is gone we start at a sub one question well the a sub one because you know you said you're bring that the variable down and leave the inside alone so where's the x- C attach to a sub one right here on that one yeah that one the Dera of that what's oneus one never mind okay you're right it happens I know I'm right right I like hearing that though still especially for my wife that never happens but uh espe she's my anyway um so here's what goes on when you take the derivative of a power series your first term is a constant it goes away so we don't have the zeroth term anymore we start at the a sub one term or when n equals 1 does that are you with me now what's happened here is for every single term that we've had we've taken the power we've moved it down but notice one 1 2 2 3 3 n n we're moving the N power forward we're leaving the a subn alone actually a sub 1 okay a sub 2 a sub a sub 1 a sub 2 a sub 3 no problem and we're doing x - c to what's going on to our power what are we doing with it do you see that this from here to here look at look at that it should look very familiar to you from here to here do you see that that is just the chain oh that's all that is it's just or the general power rule if you learned it as general power rule says hey take your take power move it down front no problem we got that um that's a coefficient leave it that inside of something chain rule leave it subtract one from your exponent no problem multiply by derivative of the inside hey it's one it's just the chain rule that's all this is it's just very cool that that still works for our power series maybe it's not that cool maybe it's obvious to you because a subn is a constant anyway uh but most wow that's kind of neat but when you think about it it should be apparent it should be well if that's a constant that's a constant time something X raised to a power you can use the chain rule with it you guys with me okay next one so we had let's call this part A derivatives Part B integrals if I do the integral of f ofx DX let's go right from here if I integrate term by term which I can do because if I can dve term by term I can integrate term by term no big deal we've done that for years now if I integrate if I integrate term by term what's this going to is it going to be a zero no integrals don't work like that if I do like one minus or plus X DX the integral of one becomes okay sure so I'd get like x + X2 2 so okay now here when I do my integration it's not just going to be a0x it's going to be a sub 0 x - c but the same idea basically happens it'd be like integrating with a u with a substitution kind of like that where you'd have 1 + x - 3 say squared or not squared you just do it like that you'd use a substitution wouldn't you you'd make u = x - 3 and you go okay well no problem then when I do an integral that would be U and when I did my U Back to my x's I'd get x - 3 same things happen in here does that make sense to you okay so we get a okay no problem what's the next one then oh come on that's yeah am I going to have an xus C what power is it going to be raised two over next one and so on and so forth which looks like this the integral of FX DX therefore would simply be a Series this one does start at zero still we get a subn we get x - c to the let's look at what happened into our power what's happening every single time we had we didn't even have an xus C now we have one to the first we had an x- C but now it's to the second that's right over what that's right does that look familiar too that's our basic integration rule it just says hey you know what what happens here uh you take this you add one to your exponent bam you divide by your new exponent got it we don't need to do worry about the x- C because when you do a substitution it's like the most trivial of substitutions the derivative x- C is one Therefore your DX is a straight substitution to the DU whatever you use for your substitution and we got it should P feel okay with this so far you shouldn't there's one thing I'm missing don't forget the plus C you see you see now uh there is one other thing that can happen you see all of these Series start out with an interval of convergence right some domain so it's like from like we had before 1/2 to 1/2 where sometimes the end points are included and sometimes they are not included now what can happen is if you do a derivative or if you do an integral of your power series you can lose end points or you can gain end points what's interesting is the interval of convergence will remain the same the interval is the same but you can lose end points on derivatives and you can gain convergence of end points on integrals does that make sense to you so it's really it's kind of an interesting idea but when you do a derivative or integral of your power series all you need to know is that the interval is not going to change but the end points might so be careful on that one show pans feel okay with the idea now I'm going to show you one example on how to use this very quickly what we're going to do and this is going to be our stepping stone into the next section I want to find a power series representation for this for Ln of 1 - x on the interval -1 to one I want to find a power series representation for that thing what we're going to do what we're going to do is we're going to use this idea to do that so here's the start are you ready for the start I'm looking for some some function that when I take a derivative or an integral of it it gives me Ln of 1 - x I'm I'll say it again I'm looking for some function that when I do a derivative or an integral it gives me this thing right there does that make sense to you now I'm going to cut to the chase because I'm running out of time I want to make sure that you get this well here's a deal I know that if I start with 1 over 1 - x and I do an integral of that thing the integral is going to be pretty darn close to that do you see that it's going to be negative but it's going to be pretty darn close are you with me on this folks so the integral of this this thing gives us negative Ln of 1 - x quick head out if you're okay with that okay now the other reason why I'm choosing this one why I'm not saying well tell me something I do a derivative for that I'm also trying to make a series representation what I know is that this thing right here this is so cool this is the sum of a what type series do you see it that's a sum of a geometric Series this is the sum of a geometric series gec power of a geometric series with okay you tell me if this is a geometric series what's my a one very good a is right here what's my r therefore this is a geometric series with n = 1 to Infinity of 1 what else do you remember how uh geometric series look geometric series look a r to the nus1 so a * R to the nus1 that's it that's my geometric series okay well that's cool or you could do it this way if you want we all see this way Nal 0 to Infinity of a x to the N either way in both of these cases in both of these cases what do the terms of your series look like what's the first term here good notice the first term here would also be one you with me so I don't care how you do e either way this one's a little easier to see okay but either way if you did that you'd probably raise to one but if you did this you'd still get one plus what's the next term come on we got to this one x to the sure plus what else plus what else plus what else and so forth and so on okay I feel like I've lost some I've left you in the dust a little bit CU I'm getting some blank looks I never like the blank looks you know you know sometimes when I speak uh Spanish to my dog he goes uh it's a little bit never mind anyway yeah this this is a little greek I I get it it's a little weird find a power series representation for that thing what we're trying to do is say hey you know what I've got a function that represents I'm sorry I have a power series that represents this function on that interval we'll talk about why the interval is there right now actually um first thing we do is say hey find me something that when I take an integral of it it gives me this thing or that thing with a positive or negative I don't care as long as it's this thing CU negatives I can deal with well I know that if I integrate this it is going to give me this what I also know is that this thing is the sum of a geometric series with AAL 1 and with Ral X look up the geometric Series this is going to be apparent to you well a geometric series is always one of these two forms no problem it's a it's a it's n = 1 1 with nus one or Z with n and it's R to that power that's exactly what we have right here so if feel okay with with that now those terms are 1 + x + X2 + x plus x 4 plus blah blah blah blah blah blah blah this how geometric series look okay now can someone out there who's thinking critically explain me why this interval is the way that it is why X must be between -1 and 1 for this of oneide kind because of absolute value of R is greater than one why why is the absolute value of R less than one or greater than one important for us conver or diver what type of series do we have so if this if this is not negative 1: one then my X which is my R would be outside of the interval of convergence does that make sense geometric series only converge from1 to 1 hey it's a geometric Series so because it's a geometric X is r h you know what r r in this case since r must be less than one for convergence of a geometric series X must be less than one for the convergence of this geometric power series I want to write that out because it's important make sure that you actually get that do you you guys notice understand the concept here the reason why this is defined only on negative 1 to one is because we are going to be using a power series to represent this we have to use a power sorry GE geometric power series to represent this because geometric series only converge when this number is between 1 and 1 it must converge when this number is between negative okay between well X has to be between 1 and 1 that's the only way that this thing is going to converge therefore it's the only way that we're going to get a function to represent this power a power series represent this function power series only represent functions where the series converge therefore we have to have an interval for which our power series is going to converge to represent this function does that make sense okay almost there very very close to almost there well what we have right now is this we got this is the sum of a power Series this power series looks like this therefore one over 1 - x is = to 1 + x + X2 + x 3r + forever never never never what I also know about equations can you guys make the jump with me what I also know about equations I can integrate both sides and since I know how to integrate a power series I can integrate term by term that's where we're making this jump you're saying hey integrate that term by term well check it out what's the integral of 1 1 - x what's it's really not it's not L it's not this it's not that it's close but it's not that it's negative do you know why you'd have a u u sub 1 - x du = DX that gives us that negative does that make sense now I'm not going to do a plus c here I am on the other side I'm not doing a plus c because very similar to our differential equation section if I did a plus c I could just subtract it and get a different constant okay so I'm not going to worry about it here I'm going to get x + x^2 / 2 + x 3r 3 plus blah blah blah blah blah and then plus C at the very end so close almost done tell me what I could do with that negative say what now if I divide by negative all these signs are going to change I'll get Ln 1 - x = - x^2 2 - x 3 3us blah blah blah blah blah minus C so close almost done oh we got to find out what C is equal to so if we want to find out what C is equal to it's pretty easy to do do you understand that every single one of these terms are going to have an X in it except for my C plug in xal Z if you do that everything goes to zero except for my C so if I let X = 0 I get Ln of 1 = C Ln of 1 = C how much is Ln of one so then 0 equals c okay so Ln of 1 - x = X - x^2 / 2 - x 3r over 3 minus blah blah blah blah but no plus C if we factor out that netive which we could have done from the very beginning we could have had Negative X + x^2 / 2 + x 3r 3 plus blah blah blah blah blah all we got to do is find a power a series representation for that thing and it will'll be done and that's really easy to do what this is can you represent that thing as a series The probably x x is going to be up there somewhere yeah uh to what over yeah do you guys see it 1 1 2 2 3 3 4 4 NN where's n start it starts at one it goes to Infinity so here's what we know we can represent this function Ln of 1 - x by this series on what interval why was1 to1 important because that's really a geometric series that's where we got this from okay we can't Define it other than that so what this says is that on negative 1:1 this thing is going to converge and it will represent this function so if you wanted to find out hey what's uh Ln of 1 - 12 you could actually plug in 12 here here here here here and do it for Infinity it's going to be exactly the same thing it's going to represent that function exactly on this interval as long as you go and to Infinity does that make sense to you this kind of out there stuff but what we just did is we took a function and we use a power series to represent that function on a given interval that for which our series converged so depends if this one makes sense to you okay any questions or comments at all before we stop all right I