in section 9.6 we're going to talk about something you see if you haven't noticed this uh we've had a lot of conditions that say well for integral test you have to have positive continuously decreasing for the comparison tests we had to have positive terms positive terms and we had to compare it for limit comparison test we had to have positive terms how do we deal with negative terms well one way we deal with negative terms is alternating series test that's when it alternates we go positive negative positive negative positive that's great no problem what if it doesn't alternate how do we deal with negative terms if your sequence from which your series is built does not alternate and that's what this comes in it says well you know what you can do you can do what's called absolute convergence I'm going to show you what absolute convergence is I'm going to tell you when you should be using it I'm going to give you two tests that are built upon the absolute convergence idea these are the some of the most powerful tests that you have okay one very very powerful test called the ratio test which we're not going to get to today we'll get to next time on this video so here's what absolute convergence says if you're given a series if you take the series and you take the absolute value of the terms so you take the absolute value of the sequence from which your series is built what that would be for instance would be a sub 1 an absolute value plus a sub 2 an absolute value plus blah blah law forever and ever and everever so basically it changes any possible negative terms into positive terms do you guys get the idea what absolute value does so absolute conversion says hey you know what consider you got a series no big deal take an absolute value of that series and it's just going to be an absolute value of your terms all of them added together is there ever going to be something you subtract is there ever going to be a negative okay if this thing is convergent so if this is convergent then we say the series that you've got this from is absolutely convergent so we go okay is a series convergent I don't know I don't know if it's convergent let's do the absolute value if we we do the absolute value and the series of the absolute value of as subn converges then this original series is called absolutely converion I'm going to try to go quickly so I get you at least one example U but let me let me show you what this is for so this is used for things like this it's used for lots of things but especially things like this which we're going to deal with this example next time series of of s of 2 n/ n^2 tell me something uh is it always positive no no is it always decreasing no tell me something is it uh is it alternating no because sign is continuous and goes like this between negative 1 and one the whole so is it alternating no there's a series of terms where it's like negative negative negative positive positive positive positive POS so is it always positive no is it alternating no it's not always positive and it's not alternating this is where we' use absolute convergence for sure it'd be very easy with this I'm going show you that probably next time so not alternating not always positive let me give you an example that we we can do we're going to do this one next time but I'll show you minute how about that one series from 1 to Infinity of 1 nus1 n 2 tell me something you know about this series you should know it because we it's alternating that's right is it uh is a limit equal to zero of that sequence from which a series is built yes and is it always decreasing yes yes it sure is we know that's convergent check out what absolute converion says it says hey how about this one how about if I just go well let's do the series of the absolute value what would an absolute value do to this right here so this would be the series of 1/ n² tell me something but by the way are you okay on doing that ABS value n s is n s no problem absolute value of1 to I don't care what it's to it's going to be pos1 tell me something about this P ser and is it a convergent P series yes so this is a p series P equal 2 p does equal two that says this converges now don't stop there say one more thing here's what this whole thing means okay check it out if you take the absolute value of your series bam your sequence basically from what you you get your series if you take the absolute value of your sequence in your series if this series converges then your original series absolutely converges absolutely as being I took the absolute value it's absolutely convergent this thing is so we say this way so series -1 n -1 / n^ 2 AB absolutely convergence now if you're wondering wait a second why is that word absolutely important why does it matter I'm going to show you with the ne next and final example that we're going to do why there's a difference between converges and absolutely converges you with me so here's what we're doing we're taking the absolute value of our sequence and we're seeing whether that new series converges if it does then we say our original series absolutely converging you know what I'm not going to have enough time for this one I'm going to write it down we'll do it next time but here it is I'll consider this I'll talk briefly about it right now but we'll we'll talk more about it next time do you recognize that as the alternating harmonic series yes alternate harmonic series is convergent we proved it with alternated series test last time how about if I take the absolute value I get one over n what's that that's harmonic series and it's Divergent you can have series that are absolutely sorry U convergent but not abs absolutely convergent which says absolute convergence is stronger than convergence and we're going to talk about that next time so what's the all right so let's continue our talk about absolute convergence now I left you with this example we talked just a little bit about it so let's consider the series from 1 to Infinity of1 nus1 / n there's something we know already from the last section that's alternating series we know that this right here is called the alternating harmonic series you guys remember talking about that it's alternating that's the harmonic Series so this was the alternating harmonic series Also let's think about alternating harmonic series firstly it's it's alternating right does the limit of a subn go to zero as n approaches Infinity yes is the limit sorry is a subn decreasing yes by the alternating series test this is for sure convergent so it converges quick head if you're okay with that so far that's old stuff all right now here's what we know we know from this section this absolute convergence that if we take the absolute value of our sequence from series is built if we take the absolute value with inside of our series and that series still converges then our original series was called absolutely convergent because it's converges with its absolute value does that make sense to you so let's go ahead and do it so if we take the absolute value of -1 to the nus one over n what does the absolute value do for us what's it going to do here yeah right so what's it doe to this Ser what's it change it to anytime you have a negative one to any power you're going to get one or negative one no problem in abs value would take the one and make it one take the negative one and make it one it's going to get rid of any of those negatives so this thing that's completely gone what do I get and the N does it change the N no now think back a little further tell me what series this is that's right tell me something about the harmonic Series yeah you could have also said it's a p series couldn't you with a p equals 1 either way what do you know about it that diverges so this is the harmonic series and it diverges now let's put this together so here's the deal if you have a series and you take the absolute value of it and the series that you get after that diverges what that says is that your original series is not absolutely convergent now here's the problem that students run into here they go oh well look at this if I take the absolute value of something and it works out to be Divergent uh my original series is Divergent is that true no no we started out with this claim right that's the alternating harmonic series we already proved it converges does that make sense so when I say okay cool well let's do the absolute convergence test well if I do absolute convergence and I get something that Di verges all that says so my original series is not absolutely converging or does not absolutely converge I'm going say it this way is not absolutely convergent okay what's what's that mean well here's the deal you can have a series that is not absolutely convergent but is converging you know what what in the world are you talking about well let's look at it we have a series here that we know for sure is convergent by the alternating s uh series test we've already shown it in the last section we had it when we do absolute convergence of that same series we get a series that is Divergent are you with me still saying that my original series while it's convergent it is not absolutely convergent because it does not pass the absolute convergence test it doesn't work so fany you understand that concept what this is called the scenario when you have a series that is not absolutely convergent but it is convergent so convergent but not absolutely convergent that's called conditionally convergent does that make sense to you so that that's the the name of it so when you have a series that converges but does not absolutely converge it's called conditionally convergent so this scenario so this this idea is called condition convergence so we got a few things going on we can have a series that's just straight up Divergent if I just gave you this one just the series on the board alone nothing else around it just a series of 1/ n that's a harmonic series that's Divergent nothing to it does that make sense to you okay we can have a series that is convergent but not absolutely convergent so here we go alternating harmonic series it's convergent it's not absolutely convergent or we can have a series that is absolutely convergent now here's the thing absolute convergence is stronger than convergence you can see it right here you can have a series that converges but is not absolutely convergent what that means is that anytime you know that a series is absolutely convergent you know for sure it is convergent absolute convergence is the strongest thing that we have so we have three levels we have Divergent nothing we got conditionally convergent that's kind of like okay cool it's not too the absolute convergence level but it's still convergent and then we have like the Super One absolute convergence so if we can show absolute convergence that's good as we can get okay that shows that it's convergent all the time absolute convergence is the strongest that we have if something converges absolutely it converges all the time show pant if you understand that one okay so make a couple little notes absolute convergence convergence come on leard is stronger than convergence other note if a series is absolutely convergent you know for sure it is convergent let me give you a real quick example right now just to illustrate this I think I gave you this example in the opening of this section but uh we we never really worked on it so here we go let's talk about this series s of 2 N over N2 okay let's look at it let's go through the whole process proc all right a whole process would be can't shoot what is the whole process what would we do first I would always do diverence test first for sure now s of 2 n/ n^2 + 1 the limit as n approaches Infinity what do you think zero it is zero yeah this is always going between what two numbers Z and that number's getting really I don't care what it is you divide something between 1 and one by Infinity you're going to get zero does that make sense passes adversion test uh is this a p series not even freaking close come on guys no how about a geometric series there's a sign in it how about uh telescoping series we haven't talked about that one in a while no how about integral test no I don't want to do that that that'd be really hard to do the integral I don't even know if we can do the integral of that one okay no it's not alternating because sign doesn't alternate just negative 1 1 negative 1 one it it oscillates between them that's true but it doesn't alternate like that so we couldn't use an alternating series so the one thing we're left with is well maybe we try absolute convergence here let's see if that that does a trick we can't use the comparison tests because our terms are not always positive does that make sense so we can't use it straight up right now but what we can do is say all right if we do the absolute value if we do the absolute value let's consider what the absolute value of sin 2N does so I'm going to take a little break here for a second uh do you agree that s of X is always between1 and 1 okay well by definition what that means is that the absolute value of sin x would always have to be less than or equal to one that's what an absolute value even means by definition okay well in fact it doesn't have to be sin x it be sign anything I don't care what you put in there as long as it's sign of something then the absolute value of sign of whatever is going to be less than or equal to one show fans feel okay with that one well we're going to change this then we go okay well cool well then what this thing does is this leads us to this statement well if the S 2N in terms of absolute value is less than one if I divide both sides by n 2 which I can do because n's positive so n squ is positive obviously then we have this statement what we know right now is that the series Let's see we know that the series of this sequence this sequence is always less than or equal to 1 n^2 what do we know about the series from n = 1 to Infinity of 1/ n² tell me something about that why are we doing that it's a p okay tell me something about the P series converges when what happens is p greater than one here so we say something about that we go okay this is a p series we say the p is two that's greater than one therefore this thing converges then we keep on going now okay we we're doing a lot of thing can you see that we're doing more than one test here we can do several tests within one problem we're doing two tests right now we're doing absolute convergence actually we did Divergence test first we're doing absolute convergence okay cool well if we do absolute convergence then we can make this claim sign's always between 1 and one therefore absolute value of sign's always less than or equal to one therefore absolute value of sign 2N is less than or equal to 1 okay well if that's true then absolute value sin 2N / n^2 is less than or equal to 1 over N2 since I know the series of 1/ n s is a p Series S of p is 2 this this series the series of this sequence here is going to converge tell me something about this series then by what okay so by the comparison test this series also converges almost done we're not quite done but we're almost done I want to see if you guys are okay getting down this far so we we start with a series that we really don't have an alternating series test for we don't have much to do here we don't have comparison test comparison test because that's not always positive we can't do it so all right we'll just take an absolute value of it we do an absolute value of this thing we start drawing some comparisons We compare this to 1 over n^ s well what that means is that because this series is series because this series can converges and our series with the absolute value of it is always less than that this series must also converge now go back to our original we're not back to our original yet so go back to our original what's it say about this s we don't stop here all You' said here is that this series converges that's true but is that your given series go back to your given series what's it say about this series then it converges use the correct terminology it can say what now converges Abol not conditionally converges absolutely absolutely it's absolutely convergent why is it actually convergent well when you took an absolute value of it the series converges therefore this is absolutely convergent does that make sense to you therefore our series that was given to us this s of 2N n^2 you always want to make some statement about your original series okay you don't want to just go down here and says uh convergence that didn't tell me anything what what are you talking about this one this one what are you talking about here we're doing several things you got to work it back to make some statement about your original series we're always trying to say whether this thing diverges or converges uh absolutely or is conditionally convergent that's the idea here so therefore here it is one more time okay did we know anything about this one by itself no we had to take an absolute value okay take an absolute value well within the absolute value we're going to be using a comparison test the comparison test said because it's a PC P less than p p greater than one and our series here with the absolute values around it is always less than this one well because this converges then this converges by the comparison test no problem now this is not the series Series this is this series with absolute values around it therefore because this series converges we say that this series is absolutely convergent show hands if that madees sense to you now absolute convergence is about the strongest thing we can do right now okay so if we say it's absolutely convergent that's the strongest we say that it's definitely it's definitely convergent so we'll say absolute convergent and then be done therefore this series is absolute abolutely convergent if I say it's absolute convergent because I had to take absolute value to show convergence here that's the strongest thing we do at this point show fans feel okay with with that one for real okay good now there we go by the way if a Series has all positive terms that absolute convergence doesn't really help us all that much it's just another test for convergence because you'd have to use something else within that what absolute convergence says is this if there's no known test that you have right now it's not alternating try it to doing the absolute value because that's the strongest thing you can do anyway but then that's going to show you possibly you could be absolute convergent all right now we have two tests that incorporate this absolute convergence this is the first one I told you about it at the opening uh the first one's called the ratio test it's one of my favorite tests it's a very powerful test we're going to look at that right now are there any questions before we we get going on that okay oh sorry ratio test first so ratio test here's the ratio test says it says if you make a ratio between your next term and the given term so basically on your sequence you'd have a subn + 1 and a subn if you make a ratio between them so if dot dot dot if the limit of the ratio between your terms is less than one or the limit between your terms is greater than one or the limit between your terms equals one after you take an absolute value of it we get three different outcomes for our series and this is what's really nice it gives you a gives you an instant picture of whether you are convergent or divergent or well whether you have to do more work on this thing so here's how it works it says what I want you to do is I want you to compare the sequence from where you're getting your Series right here's your SE sequence here's the a subn term here's the a subn plus1 term it's very easy to find this I'll show you exactly how to do it take that that comparison take that ratio please make sure you have it written correctly do not invert these two terms otherwise these are obviously going to be all wrong okay so make sure you have your next term over your current term your a subn plus one term over your a subn term in every single case if your ratio after you have done the ratio and done the absolute value and taken the limit if that limit exists and if it's less than one if that's now it's not going to be negative explain to me why it's not going to be negative absolutely you're absolutely right so if this is if it's less than one what we immediately know is that our series I'm not going to do that our series is absolute conversion if we have this limit of the absolute value of the ratio between these two terms and it's greater than one we know that our series is Divergent if we have the limit of the absolute value of these two terms as a ratio and it equals one it's inconclusive and then you'd have to try hopefully some other technique that that you haven't done already so this is the idea it's not hard to do it's really nice do you guys get the idea before we get going any further it's really straightforward honestly uh you do the a subn plus one term over the a subn term put absolute values around it that's what's given us the absolute part of this absolute convergence or straight up Divergent no problem or it's inconclusive which we'd have to do something else for let me give you uh let me give you an example just to illustrate this right now last chance for were any questions on on this one okay so here we go you know many of you might have noticed something right away with uh with this right here especially this one do you always need to do the Divergence test if you're going to do the ratio test you can if you want to but sometimes it gets pretty difficult to do especially with like factorials and powers of N and all sorts of stuff so what's nice about the ratio test is that if you can't do the Divergence test right away you remember the Divergence test right you do the limit if it equals z you keep going um if you can't do that or if it's going to be really hard and you're doing the ratio test anyway skip it say I'm doing the ratio test it's going to show up as divergent if I do that does that make sense to you that's kind of nice here so it's a really powerful Test shows absolute convergence or Divergence this is the one that sucks though you do all the work you get one you go oh man I got to do something else anyway so that would be the issue let's start off with this one uh tell me something about that let's let's go through all of them okay what's the first thing that you would do here you would do you good because even with the the alternating series test that's part of it right so if I do this one is it is the limit of this sequence here is that equal to zero try some lals in your head this one's going to go down a power then down a power and then constant right this one's not this is going to be 2 to the n and then well Ln 2 2 to the n and then Ln 2^2 2 the N so this one's not going to decrease like that therefore this limit would equal zero does that make sense to you also uh well it's not a p series it's not a harmonic series it's not geometric it is alternating do you see it we already sh showed that the limit of that thing goes to zero is it decreasing oh yeah so right now if you wanted to you could use the alternating series test and show this is convergent show up hands if you you understood that you could show that right now if you wanted to I want to show you the the ratio test just illustrate on something that's a little simple so here's what we're going to do okay let's let's try to use the ratio test here what the ratio test does is it sets you up with the limit and puts absolute value around everything that's where we're going to get our absolute convergence here by the way do you understand that we might try to show absolute convergence with the ratio test rather than just convergence with alternating series test what's stronger absolute so if we show absolute convergence it for sure converges anyway we show it inherently with the alternating series test does the alternating series test show absolute convergence the first example today that we did said no it can show convergence but it can't show absolute convergence because some things aren't absolute convergence are converg in Alterna Series so here we go we're going to try to show absolute convergence with the ratio test the ratio test works this way it says what I want you to do is take this sequence from which we're getting our series and put an n + one everywhere if we put an n + 1 everywhere we get -1 to the N do you see where I'm getting n from n + 1 - one is going to be n we're going to get n + 1 2 + 1 over 2 n + 1 okay I really do need to show I feel okay with that so far that's about half of us are there any questions do you see where this is coming from one more time okay what we're trying to do is we're trying to get this right now this a subn +1 term the a subn plus1 term means hey go to your a subn plug in N plus1 so instead of n I'm going to have n plus1 instead of n I'm going to have n plus one instead of n I'm going to have n + one now n + 1 - one is n question um couldn't you just drop the negative one in since it's absolute value we're going to talk about that next yeah it's a great idea and you can after you show me the absolute value okay so right now no you got to have it somewhere um but now what here's what you could do all right if you wanted to right off the bat we just talked about absolute convergence right and this is going to show us with the ratio test absolute convergence do you guys get me so right off the bat you could just go well let's do this let's do absolute value let's get rid of that and then do a ratio test it's going to work out exactly the same because all the absolute value does for you is get rid of the negatives doesn't do anything else so depends of that made sense so either way you want to do it I don't care the ratio test practically puts the absolute value in there anyway U so it kind of does it all in one fail swoo all righty that's a good question thanks for that one now over it says put that over the original sequence anyway well that's this thing so all you got to do is rewrite that so here's the main point okay do not invert this or reciprocate this fraction it's a complex fraction this always goes on your denominator don't put it on the numerator this thing just rewrite right here so -1 n -1 n^2 + 1 okay all over 2 to the N you know as a matter of fact um I kind of wrote my fraction poorly this really is one times the whole thing so if you want to be precise about it the way I wrote it you really should have some parentheses there uh that way we're not algebraically incorrect it's not going to matter for the next thing we're going to do here's why it's not going to matter for the next thing we do um like Devon said what's the absolute value going to do to this gone absolute value negative one or one is going to be one this thing gone as well so that's what's nice about the absolute value and like I said you can do it first right off the bat and get rid of it and then do the ratio test that's fine fine so we're going to have a limit and approaches Infinity I'm going to do a couple things here first thing I'm going to do is I'm going to get rid of my negative-1 to the NS second thing I'm going to do I'm going to take my complex fraction so ignore these okay these are going to be gone I'm going to take my complex fraction I'm going to multiply by the reciprocal is that okay with you yeah so we have n +1 s + 1 over 2 n + 1 * 2 N over n^2 + 1 this and then the reciprocal question true or false I can get rid of my absolute value right now true or false no false okay why is it false you can still have a negative up in there somewhere can I still have negative pop up in there somewhere where's n start is this ever going to be NE so if you want to put them here that's fine but are you ever going to get a negative here no here no here no do you need absolute value no no okay so at some point you should be able to drop those absolute values are you with me at some point does it make sense to you I know we're doing a lot here but I want to make sure that you are with me are there any questions on that at all so do you show up hands if you feel okay about where this came from a ratio test are you okay that negative ones they're gone absolute value hey gone that's why we're doing this to get rid of that stuff no problem uh reciprocate by multiply by reciprocal not a big deal can we simplify this thing yeah first The Next Step you know I'm going to get rid of the absolute value not an issue we just talked about that what simplifies here two the two to the end simplify you need to know this is going to happen an awful lot on your ratio test an awful lot you see when you have this 2 the N plus one you're going to get a lot of plus ones aren't you well understand that this is really 2 the n * 2 that's really what that is does that make sense to you so instead of instead of this one I don't want I don't want that I'm look at this one the two to the ends are gone those simplify what I'm left with is just two does that make sense do you understand that 2 the n + 1 is 2 the n * two because when you multiply common bases you add exponents and that has a power one right there this is this you can simplify two to the end with a two to the end and then we have just two left just a little algebra but I want make sure you see it let's keep going well that means we got a limit as n approaches Infinity I'm going to drop the absolute value cuz everything's positive anyway it's not really relevant for us I'm going to distribute this thing this will be n^2 + 2 n + 2 over this will be 2 n^2 + 2 I want I want you to verify the algebra on your own make sure that I'm right on this thing am I right you sure you're okay with that one so this sayy that's two no problem this is a great simplification for us we distribute that you get 2 n+ 2 this one distribute no problem combine some like terms there and we got that now tell me something about oh my gosh this is an easy limit this is a good one what's this limit perfect why is it 1/2 I don't care how you do it you do L Call's rule you can do whatever you want with this uh that's going to be 1/2 you can divide by large powerometer understand that limits are always about leading terms you get have whatever you want to do that's not what I'm trying to teach you here what I'm trying to teach you is what's that result say go back to your ratio test if the ratio if the limit of the absolute value of the ratio is less than one or greater than one or equal to one we have three different results the worst one is inconclusive right these ones I don't we really don't care what we have as long as we have one answer right it fits one of these models what does that tell you about our original series absolutely conver it's absolutely conversion that's right if that limit is less than one we know our series is absolute conversion so what we're going to say is so by the ratio test are you getting the picture that whenever I tell some whenever I say something I'm saying buy this by that blah blah blah blah blah we're we're stating what we used we're saying by the comparison test because this is a p series it's absolute conversion because we did the ratio test here and it showed this and this was less than one by the ratio test the series from one to Infinity of whatever I gave you to start with is absolutely convergent and that's really nice that's really it's a nice I don't say easy test there's a lot there's a lot to it right but it's definitely not that hard if you know how to use it uh would you like to see at least let's do like two more examples you guys good with that yeah let's do some examples then any questions is it straightforward enough for you question this kind of be like our goto test for most oh yeah I love this test now you I'm I'm not going to say go to because sometimes I mean they're so easy that you just like a telescoping no all right or like a geometric that you can see it's geometric no P series you see p series obviously not no alternating series some yeah you might do it for that um I'm going to give you some hints at the last part of this lesson it's not going to be today but I'm going to go through like when you should use what I'm going to do a complete recap of all this stuff and that's going to be kind of nice for you uh when you use the ratio test typically here's a little preview when you have factorials especially factorials it's like number one okay factorials and when you have powers of of n that's usually when you do this is this the only way to prove uh absolute convergence or taking the limit of no we did it without this so that can that prove it though is that a test can you you can prove well absolute conversion simply means when I take the absolute value of a series and it converges my verion series absolute conversion the ratio incorporates that into it which is why when I do the absolute value it's not just converion oh it's absolutely okay so let's go at the series from one to Infinity of n Over N to the N just joking insert a little humor there uh now let's look at it all right let's look at this first thing I want you to notice tell me something about the Divergence test would you want to do that here no I wouldn't either because n factorial of n then I don't I don't know man it's probably zero I guarantee it is because this thing is going to I think it's converges uh this is going to converge but would we want to show it probably not uh which right there you go let me do the ratio test why why would I want to do the ratio test here factorial factorials and it's so nice with the ratio test you're going to see why it's nice in just a minute let's just look at other things real quick is it a p series geometric telescoping no no uh harmonic no is it something that I would want to do an integral test on no no comparison test maybe we could possibly U here's why I say that are they all positive terms yes MH they are which means the absolute convergence well it's just going to be another test for convergence for us what's great though the ratio test doesn't care about that the ratio test just says hey I'm going to tell you whether you're ABS convergent or Divergent you don't need to do the Divergence test the ratio test so it'll be in there somewhere does that make sense so that's kind of cool let's do the ratio test and see how this thing plays out so ratio test said and you can even write by ratio test so ratio test says do a limit as n approaches Infinity set up an absolute value even though right now we know it's all going to be positive okay we still set up the absolute value because the ratio test does at least showed on the first step after that yeah they're positive terms get rid of it no big deal let's set it up first so so you get in the habit of it it would stck to you know get in the habit of not writing it and then need it here and oh man you don't want to do that now let's see if you guys can fill out the numerator of our complex fraction and the denominator so I'll give you about 10 seconds see if you can get here and here go for it the denominator should be really easy because the denominator is exactly what this says that's it that's our a subn the numerator takes a little more work but not much this is going to be n + 1 factorial over n + 1 use parentheses where appropriate to the n + one show hands if you got exactly that so everywhere you see an N you should put n plus one that's that's all it is now let's talk about this for real are the absolute values relevant is that positive is that positive then this whole thing is so let's drop the absolute value also with the ratio test you're always going to get a complex fraction you're typically going to get a complex fraction usually with us um so if we get a complex fraction we're going to create a division problem and multiply by the reciprocal to undo that complex fraction so we'll have n +1 factorial Over N +1 to the N +1 times what's going to be next did I tell you you guys are going to get a lot of n plus1 Powers what typically happens here is you'll have to split off one of those powers like we did with the two like we're going to do with this thing so don't let this confuse you we're just going to split one off just break it off strip it off there oops so we got limit no problem we're going to have an N +1 factorial we're going to have an N to the N we're going to have an N factorial and then we're going to strip off 1 power of n + 1 we're going to strip one of those things off so we're going to put n + one to the n and we're going to strip one off we're going put that little stripper right here n + one oh come on that was funny just joking do you understand that this really is n +1 to the n + one power yeah this is a power one right common bases if you multiply common bases you add exponents that is this thing okay well that's not not bad that'll help us out in just a little bit so what now oh goodness gracious simplify can we simplify that do you understand what n +1 factorial is do you know that n +1 factorial is the same thing as n +1 time what's oh what's the next term n n oh times this would be n minus one oops all the way down to one does that make sense to you tell me what this is therefore by your own little log here not little I don't mean to mean you by your own logic uh this is the same thing as n +1 * n factorial let's see if you can wrap your brains around that does that make sense to you if this means n times all everything previous to it and I multiply by n + 1 that gives me n +1 factorial this is n +1 * n factorial n+ 1 * n factorial that is the idea show fans you feel okay with that one get used to it you're going to do that a lot okay so let's uh let's go here so what I'm going to do is I'm write the limit and we're going to have n + 1 * n factorial over this guy we had n + 1 to the n * n + 1 this guy was n to the N this is n factorial do you see any things that can simplify now by the way everything we're doing right here is algebra there's no calculus involved right now it's all algebra it's understanding that this is a complex Frac in fact even doing this this is the only calculus you really really do besides taking a limit of the very end you're going to do some some it's not going to be that easy uh to do the some of the limits but calculus here all algebra to simplify that's really it understanding that that's a complex fraction no problem understanding this right here is n + 1 * n factorial that's a big deal understanding this is n +1 to the n * n +1 okay these two don't change at all let's see what we can simplify give me something n factorial is gone oh that's a huge deal those are gone what else n+ 1 that was important too so what we end up with is a limit as n approaches Infinity of n the N over n + 1 to the N you see the idea here that I was trying to get to was two terms which have the same power did I get that simplify as much as I can I got that well then if they do have the same power I can get n/ n + 1 to the end we're almost home almost done now this is where it gets a little weird right um I want you to think of this for a second I want to open your memory banks here right now folks hopefully you have some cash to withdraw from otherwise your memory banks are whatever um plug in Infinity what do you get no you don't what do you get in here L than one you get one this is one this is infinity you get one to the infinity that's an indeterminate form remember indeterminate forms when you had a one to Infinity oh this is this is this this is where we went oh let's do e to the limit of Ln of bunch of junk do you remember that and then we did a bunch of junk because you move your power forward you create a fraction you use lal's rule all right you're going to have to do things like this now I'll give you a couple hints most of the time what we like are fractions that you can split apart we don't like n Over N plus one we like n plus one over n so when we do that maybe do a couple nice little tricks here do something like this do the n + 1/ n now does anyone know about this if you reciprocate a fraction what happens to that becomes negative this becomes negative n which means I can write this as 1/ n + 1 / n to the N now I'm GNA cheat okay I'm not cheating I already I did on the side uh I'm not going to reshow you this this limit from n to Infinity of n + 1/ n to the N this do I have time do you guys want to see it be like a minute two minutes over U but here it is firstly do you understand it's one to the infinity okay that's bad news what that means is we do e limit n approaches Infinity of bunch of junk to the N with an Ln in front of it that allows us to move our n forward for the sake of time I'm going to not write uh just that little part when we have that this is now an infinity times sorry that's Ln times one okay so we're going to have to redo that one so that's e to the Limit Ln of n + 1 n all over 1 n now we can do a ly talls with this thing so if we do a ly t with this we're going to have e to the Li 1 n + 1 n * -1 n 2 all over1 N2 the Der of this is this the Der of this is this you're going to have to follow me care are you guys with me on that one I know I'm doing it fast well this is kind of nice cuz those are completely gone so this is e to the limit of 1 over if I divide both these I get 1 + 1 in remember that n is approaching Infinity what's one over infinity 1+ 0 1 over 1 e to the 1 that limit is e therefore this thing right here so this was just its own little own little guy what that says is that the limit I didn't mean I didn't really want to show that but now that you have it I'm sorry what's the limit of this thing this is one Reas now tell me something about one over e that is less than one this is less than one and so we're going to finish this thing off um what what test did we do let's talk about our original series tell me something about the ratio test was it conclusive did we get anything out of it when is the ratio test not conclusive very good this thing wasn't equal to one therefore we have some conclusion uh the fraction the ratio was it less than one or greater than one what's it tell you about our series you're absolutely right what I want to know is if that makes sense to you okay let me ask you a different question did it make sense up to this part show hands if it did okay were you able to follow that one that's old stuff all right so you got to this is uh the section where we did the indeterminate of limits you might have to do that Within part of it understand you can take a piece of this do the limit put it back and then follow up with that ratio test you guys okay with that with one more example then I'll do a recap next time all right here we are so let's do our last example of using the ratio test uh then we'll talk about the root test and finally a recap of everything that we've done in our series thus far so getting started uh first things first what's the first thing that you would probably do here alter you could try lots of things here this this is going to alternate for sure um I probably might just jump straight to the ratio test because we have that negative up there because we know it's nice and Because unless it's inconclusive it gets gives us the result that we want either divergent or convergent you see some of those other tests they don't they don't do that uh some of them are just well if it does this then it's this this one says well we have three outcomes we have absolutely convergent which means it's definitely convergent we have Divergent which you're done anyway or it's inconclusive if it's if you get listen on your homework and stuff they're going to ask you for a couple things right they're going to ask you for whether this series is absolutely convergent conditionally convergent or Divergent does that make sense if you want to check for absolute convergence you have to do an absolute value which means you're probably going to use the ratio test at some point so how you go ahead and do those those problems you start with a ratio test or the root test which we're going to use you start with that if it works out to hey it's less than one you know you're absolutely convergent you're done that's stronger than convergence if it works out to fraction greater than one no problem you're done it's Divergent no need to go any further if it works out to one then you start using another test after that question now what if it's Infinity that's over one that would be over one yeah but or if it's zero so it counts as absolutely okay yeah okay so let's uh let's start here let's try the ratio test on this thing so first thing with the ratio test is what what do we do with the ratio test we look at me blankly it's a good start uh what do we do with the ratio test how do we start that thing off okay it's a limit that's very good so we got a limit a limit as of what that's right in fact I think I told you this last time if you wanted to could you do the absolute value here and then compare the ratio yeah the absolute values are going to take all those negatives away anyway make all positive terms so you could do that as well most people just do it all in one step the way that I'm going to do it uh by the way what goes on the main numerator of this complex fraction we're creating is it the a subn or the a subn plus one so we're going to write that so a sub I'm a little crooked here a sub n + 1 and we're going to be real careful about it what what's our a subn + 1 how would that look okay to the what very good and then what and then what perfect we'll end those absolute values on the main denominator it's just a subn again so it's -5 to the nus1 it's N2 and 3 n now what would you do if this was your problem on a test what would you do here remove the abute remove how would I remove the absolute value how would I do that sure so look at you can do that actually right now if you really want to if you did the absolute value of5 to the N you're going to get 5 to the end what you can't do is just say well when I take the absolute value that just goes away like my-1 to the end that that you can't do you guys see the reason why so the reason why this actually works is5 to the end is the same thing as -1 * 5 the N which the same thing as-1 to n * 5 n which means that yeah okay you can get rid of the negative but you can't get rid of the five to the end are you sure you're okay with that okay so we can do that now as well also what I want to leave it as this complex fraction I did is that oh no so I'm going to drop the absolute value that means I'm going to get rid of my negatives I am not going to get rid of the N or the five I'm not going to get rid of the nus one or the five I need those things and I'm also going to instead of having the complex fraction we'll have one fraction times the reciprocal of our other fraction so here we'll have 5 to n over n + 1^ 2 * 3 n + 1 * what's going to be on our numerator here all over perfect yeah exactly right why would it be five your absolute our absolute value would take care of all that stuff so no more negatives what now stop simp simplify simplify as much as we can tell me some things that simplify yeah okay so if we have 5 to the nus one we have 5 to the N how much do we have left and where does it go do you understand that what 5 to the N really is is 5 N - 1 * * 5 do you guys get that so if this is 5 n -1 * 5 then the 5 n -1 would simplify and we would have five left are you sure you're okay with that it's just algebra but I want to make sure that you're you're all right yes no guys over here are you okay with that one okay moving on what else what else simplifies come on quickly we got to move on this stuff what else the threes okay so we got 3 to the N we got 3 the n + 1 understand that 3 to the n+ + 1 is really 3 n * 3 well if this is 3 n * 3 what can I simplify out of that three to the ends are gone what's remaining three okay I'm going to move over here just so I can write this a little bit nicer so we have a limit as n approaches Infinity of let's check this out we've got well first we got a five and we got a three so we got a 5/3 we've got an n^2 over n + 1 2 quick show hands if you're okay with that one so far okay next up notice how the n and n plus one both have the same power let's make them one fraction to that power so we'll have a limit n approaches Infinity of 5/3 * n / n + 1 squared hey tell me what's going to happen here what would you get what happens to our our limit as n approaches to Infinity we get one the whole thing's one let's let's go part by part okay how much is the limit of n/ n + one that's one do you understand why that's one you could divide everything by n if you wanted you could do low talls on the inside of that if you wanted either way that's going to be one how much is 1 squar 1 s is 1 * 5/3 is 5 that's a horrible 5/3 now tell me something if the limit of the absolute value of the ratio between my N subn A subn + 1 and my a subn term equals 53 is that a conclusive result for this ratio test is it conclusive or inconclusive inclusive it's definitely conclusive it tells us something now you got to know what it tells us inconclusive would be when that equals to yeah then you got to do something else all right you don't know anything about that but this one equals 5/3 is that bigger than one or less than one what's that tell you about our series that we got I don't care about absolute de there no such thing it's just Divergent okay I don't want you to say absolutely Divergent because that's that's that sounds weird all right um if you say absolutely Divergent you could mean it's not absolutely listen care it sounds trivial but it's not okay if you say it's absolutely Divergent that's kind of weird because well does that mean that it's just simply not absolutely convergent but it might be convergent that's called conditionally convergent or do you mean that it's Divergent altogether there's there's no absolutely Divergent there's absolutely convergent there's conditionally convergent where it does not absolutely converge but does converge and there's Divergent so according let's say properly what's this one do according to what test so we say Hey by the ratio test test the series that we had to start with Divergence and that is the whole thing for the ratio test show feel okay with the ratio test okay our last are there any questions at all before we go is it pretty straightforward to you question just want you to go over that how the how that 5 n- 1 * 5 equal 5 the N like I'm so how much what's the power here one add the powers oh because of the absolute the negative disar n if you add 1 plus nus one you get n okay last test last one aren't you excited I'm kind of excited the last one we've done so many of them the last one last one we're going to tr least is called the root test use the root test if you have nth powers in your series now we talked about the ratio test right I said you use that ratio test if you have factorials or nth powers in your Series so what that means is there's like a crossover a lot of times you can not a I don't know some of the times you can use both of these tests either one of them you can use a ratio test or you can use a root test it just depends on which one's going to be easier for you what I would not recommend is using the root test if you have factorials use the ratio test for that so when the root test is typically used is for a series involving n powers what's really nice about the root test is that after you've covered the ratio test the outcomes are very straightforward so here's what we do if we take a limit starts up exactly the same way as the ratio test we're star with a limit if we take our our Ser well our sequence from which we get our series and we take an absolute value of it do you notice something that's different about this test versus the ratio test already what's different about it is there a fraction at all there is no ratio because this is not a ratio test all right this is this is a root test so instead of creating a ratio we're going to create a root root so here's how it works we take the sequence itself the the main advantage here is that you don't have to have a really large well I erased the board but a really large fraction out of this one there there's no such thing it's just the the sequence from where we get our series the ace sub n then you take an absolute value of it and then you take an nth root of it what's nice is that if this thing exists I'm try to make this a little clearer if that limit exists so we start with our sequence no problem take an absolute value of it no problem take an nth root of it and then we take a limit as n approach Infinity if this limit exists and is less than one it has the same result as if the ratio test were less than one what's it going to tell you about our series absolutely that's very good it's absolutely convergent what about if the limit is greater than one what do you think yeah what about if the limit equals one what do you think same results you guys okay with the root test probably not yet right you want to see an example let's do an example here real fast just one of them to give you an idea about how you can do this again I'll make the statement one more time can you use the ratio test for sometime you could use a root test can use the root test or sometimes you could use the ratio test yeah you can a lot there's a lot of crossover there right again what I wouldn't recommend don't try this if you get factorials it the ratio test is really nice for factorials because when you just add one to a factorial most of that stuff is gone make sense so keep that in mind let's try one example just to show you how it works and then I'll do that recap I've been promising you so let's start with this series you know a lot of times students have questions how do you know what test to use what do you start with well you start with firstly you know what I'd start with through all our process and see what's going to be the easiest thing for us to do in each case um I know that's alternating do you guys know that's alternating yes is it going to be easy to show that it is decreasing or to show that the limit equals zero no those aren't going to be easy things to do so I probably wouldn't do an alternating series it's not a p series it's not a geometric series it's not a telescoping Series so the first thing we come to is like well all right uh integral test no it's not I don't no that's not all positive terms you can't even do it compar test no it's not all positive terms you can't even do it so you what you could do you could take the absolute value of the whole thing and then try one of those couldn't check you can do that and then try a comparison test then try integral test but those things look pretty hard for this one because we have to show decreasing for both of them for integral test and comparison test that wouldn't be a good sorry uh for integral test and for alternating series test you have to show decreasing comparison test well you'd have to figure out what this is smaller than then show convergence or bigger than then show Divergence probably the quickest way to go about one of these ones because it has the absolute value already is the ratio test or the root test you with me and I believe both of them would work um but I'm going to show you why you Pro you might want to choose I don't know I haven't tried the ratio test on that one maybe you can try to and see if it actually does work uh maybe I'll do that later too I would try the root test and here's why I try the root test what I'm seeing is that I have this to a power N I have this to kind of a power N I can make it a power n so if I take an nth root those things are going to go away the absolute value is going to make this thing go away does that make sense so let me show you how the root test Works root test says hey take simple limit n goes to infinity and nth root of the absolute value of whatever you have here you don't need to do an N plus one which is kind of nice you don't have to create a ratio or do anything or change this sequence where we get our series from in any way shape or form and already we can start working with it so let's go through it one time firstly show up hands if you're okay getting from here to here using the root test are you okay with where that's coming from it's nth root it's absolute value it's the same exact thing you got now let's work on it uh absolute value of negative 1 to any power what's going to happen with this thing that's gone completely gone so we got a limit and approach Infinity yeah it's still an nth root but this the absolute value of1 to any power is one so this piece is gone now 2 n + 3 when you're doing the root test you're trying to match up your powers so if I have 2 n + 3 I'm not going to write that as 2 n+ 3 again what I'm going to write is 2 3 * 2 to the N are you okay on algebra with that mm that still 2 the n + 3 that's exactly what that is by the way um if you would have had like 2 the nus 3 2 N minus 3 you can change this to 2 the N that's not even that's not even n come on Leonard wor it times 2 the -3 couldn't you do that yes that would just go to the denominator and be 2 the 3r so don't get stuck on this and go oh my gosh I can't do it well yeah you can just separate off that negative exponent move it to a denominator and then it's that simple does that make sense to you so don't get all hung up on that all over n + 1 to the N now let's try to make this all have one power if we can so here we get 2 the N we get n+ 1 the end we get 23 23 I'm not worried about here's why I'm not worried about that I can separate products when they're inside of any sort of a root so I can separate this as 2 3 * 2 n + one to the N furthermore because I've separated this I can make this a limit as n approaches Infinity of the nth root of 2 3r time the nth root of all this jump 2 n + 1 to the N power now here's what I'm going to ask you do you see why I tried to get these two things to have the same power do you see why the idea is if I can make them have the same power I can do this with it what's the nth root out of something raised to the nth power what's going to happen there they're gone so we get a limit as n approaches Infinity of by the way do you know this I know you know it um when you have a a root out of something ra to a certain power we get the base to the power over the root did you know that yes times 2 over n +1 let's consider that let's see about n going to Infinity uh so you tell me what's 3 over n when n goes to Infinity can we do that Z what's two to the Z okay so this is one times uh what's two oh how about that one what's what's n as n to Infinity this whole okay the whole thing you guys are better than that aren't you two over this what is that zero what's 1 time Z okay is that conclusive or inconclusive when is the only time a ratio or root test is inconclusive is this equal to one then there's a conclusion here uh is this greater than one or less than one hopefully yeah so what does that tell you about this series is this series absolute convergent inconclusive or Divergent what is it Con very good so we say it by the root test we tell me what we're doing our original series here is one more time if the result you get is less than one what do we say about our series absolutely that's exactly right quick show hands you feel okay with with that so far are you all right with that one so we go all right well what would work best here well probably not anything we've done before what's quickest what's nicest is a ratio test or a root test it'll give us a nice conclusive answer most of the time if it's one then you try something else in this case you go well what's going to be better for us a ratio test a root test root test probably because I have an n and an N I can make those have the same power I know that the root is eventually going to get rid of that and will have no problems with this results are the same for the root test as they are for the ratio test you sure you're okay with it now we've come a long way haven't we we've done a lot of stuff on this idea of series what I'd like to do now is give you a summary of everything we've done and when to use it that way when you get to your test or your homework or whatever you're doing man what am I supposed to do here maybe you can break this down like a little outline or or have this in your head what you should look for to make the problems as easy as possible any questions or comments or anything before we going on that would you like that little summary great like please well typically a book will have something like this in it I think ours does too uh but I would like to give you kind of my version of it as well things that you should be looking for so here's a summary the number one thing we look for in any series number one thing we always test for in any series is if we can it's nice and easy we do the Divergence test we kind of skipped over that one on this one because what's nice about the ratio and the root test is it almost inherently does that because when you get down to here if you have something greater than one you know it dages anyway so it kind of does that for you which is nice but most of the time almost all the time we're going to do that Divergence test first so number one thing we check for is if the limit of a subn does not equal zero the series of a subn diverges this is what we call the Divergence test take a limit of that sequence from which we get our series and if that limit of that a subn does not equal zero you're done it differes number two we can also do series what I call what's matter inity say what what did I do Sho you know what I I thought about that zero I'm sorry mind was ahead of my hands so yeah you're right if n approaches Infinity that's right and a sub the limit of a sub n does not equal to zero our series diverges next one we got several types of series we dealt with so we can do this by type for instance if you know it's geometric yeah I'll just do it that so if we know it's geometric of that form remember this could be to the nth power as well when do this converge when when absolute value R is greater than one between negative 1 and one so the absolute value of R would be so if we know it's geometric stop don't do any ratio tests or root tests or anything harder than that it's geometric man make it geometric manipulate to fit this thing then we know it converges if your R AB bar is less than one also what's nice about geometric is there's a sum built into it the sum is a over 1 R it's a sum for us what else do we got oh yeah when does it diverge thanks a lot for the help on that one guys more than or at least one by the way if you check for should you do the sum first on a geometric or should you check for convergence or Divergence first convergence diver if the geometric series diverges would you want to do this on a test I will Mark you down for that you say the sum is this no it's not it diverges so you can't tell the sum is okay don't do that okay next one um we had telescoping series for telescoping series the basic idea here is use partial fractions to get some s remember the partial fractions we did use partial fractions to get a a a partial sh so we start with partial fractions to get S of n that's our partial sum and then we take a limit of N S Sub n as n approaches Infinity we go okay here's here's what we have now now let's say this n goes to Infinity so we say here's our sum to a certain point move that point to infinity and we get whether this telescoping series converges or or diverges and we get a sum for of course I'm not reteaching all this we've already covered all this stuff I'm just kind of recapping okay okay last one by type last one by type the last type of series that we had was what oh no I'm sorry U kind of kind of different type like a named type like we had geometric we had telescope we had P series that's right so P series if we have 1 over n the P that's a p series do you remember when a p series converges and when the P Series dages yeah it's easy to get these confused for some people who go oh man geometric great lesson one oh what what happens it's different for this one so a p series converges if I'm sorry what when do you say it converges if p is what and when does it diverge l or equal to that's what I was looking for where's the equal to go okay so what's one what's that going to be the equal to is it going to converge or aage that would be diverges you're exactly right if I have the equal to one that gives me the harmonic series we've already talked about how that diverges anything else greater than one well it's going to converge less than one it's going to diverge also we talked about this a while ago I'm not going to write down but you need to know that a lot of times we can manipulate our series to fit in one of these so all of a sudden you know we might not have it immediately but we work on it a little bit we we end up getting a geometric or we end up getting a telescope and we end up getting a a p series and then if it's by type man this is really nice because it's super fast and for at least two of these we can find the sum for it the other ones we can't but for these two we can actually get the sum you okay with that so far yes so if you've done the diverence test and it passes and if it's not a specific type so it's not a geometric and it's not telescoping and it's not a p series the next thing you might want to look for or maybe some people skip over this one because they see well probably go to ratio test you might try the integral test integral test says that if you have a function I'm going to use some poor notation but I'm just going to do it anyway if you have a function that represents your sequence from where we get our series and the function is positive continuous and decreasing on an interval from 1 to Infinity actually you know what um this is a little bit of a not a mistake but kind of we don't need it from one to Infinity if you remember this we can add and subract the finite number of terms to our Series so even if it's not till like 3 to Infinity we had a couple of those problems right where it was okay started at three that's fine two it just has to be positive continuous and decreasing on some interval forever starting at a certain point some big n not little n but if we have that if we have that we can model our a function after some sequence we read our series and the function is positive continuous and decreasing on a certain interval then we can do the intergral test from wherever your interval starts to Infinity of course that will be an improper integral because we have one to Infinity here and what this says is that if this thing converges series converges if this thing diverges Series diverges so function positive continuous decreasing on an interval not a problem do an inter integral test from one wherever your interval starts to Infinity it's an improper integral if you find out that equals a number cool hey converges if it doesn't well hey diverges not a problem number four if we have a sequence a series from where all our terms are positive and the series acts like a p series or geometric or something else that we can do uh like a generally geometric P series or while harmonic is a p Series so if it acts like one we do a comparison test or limit comparison test we say hey this is smaller always smaller than something or it looks like another series kind of it has maybe missing one little piece to it we can do a comparison test or a limit comparison test for so when we did that we knew that if we could find another Ser another sequence really where every term of our given series was less than the known series if we if we know that and the series of B subn converges then the series of as suban also come if a sub's bigger than b sub and a series of B suban diverges Ser series of asan also Di and lastly this of course would be for the comparison test so comparison test says hey you know what if you know this if you know that every single term in your a subn is less than every single term in your B sub and the series of B sub converges as sub converges if this every term in as suben is bigger than or equal to every term in B subn and B subn diverges a sub is bigger than that it has to diverge this is for the compar test the last one's a limit comparison test says hey you know what if you take a limit of a subn b subn don't get this confused the ratio test this is not the same thing uh the limit comparison test does this in with two different series here that look a lot alike okay so if this thing exists naturally uh these would both be positive terms okay so your limit would have to be positive as well if this thing exists then the series will have same result both convergent both Divergent okay I'm going to keep moving but you guys have any questions on this I'm going have to race this as we go unfortunately I'm I'm short on wall space so are we okay with with this hello yes know should shouldn't be new stuff really it should be old stuff but I want to make sure that you guys are okay on what I've written down which one do we do conversion to the soap number five okay number five for a series where we have -1 to n or1 to nus1 times some other sequence that has all positive terms that's your alternating series you know what this one should be pretty fresh you guys should tell me this is a two-part test part A what's part a li limit limit of what a and everybody what's it got to be equal to Z yeah that's practically the Divergence test again just part of it so limit got to be zero also Part B what else has to happen got to be decreasing that's right so the derivative of a function that's modeled after your sequence has to be negative negative mean decreasing or if you show this the next term is less than or equal to the previous term in other words if it's decreasing then the series converges so fro good on these ones the last two we just did so I'm not going to be kind of super explicit on the last two what I'm going to tell you is that on number six if you want to do ratio test here's when you should try the ratio test try the ratio test for factorials and N Powers by the way uh you guys know what the ratio test is right I don't want to kind of recap that we just did it it's is it a subn over a subn plus 1 or a subn plus 1 over a subn and then we take an absolute value this will show you absolute convergence Divergence or inconclusive when is this going to be absolutely convergent when is L one when the ratio is when the limit of the ratio is less than one that's right so and if it's greater than one and if it's equal to one you have that in your notes somewhere right should be previously on the video actually number seven try the root test for n Powers as I said before sometimes you can do either one of these tests and get the same result out of it now what I'd like to do for the rest of our time here I'm going to briefly I mean like so fast go through some of these problems just to give you an idea about how I would look at them and say oh you know what I'd try this here I would try this here we're not going to do hardly any any of them okay we're just going to look at it does that make sense to you y so let me write them all down and then we'll go through them very fast for for for okay there we go I want to give the rundown here because I think it's really important that you guys see this at least one more time do you have them all written down yes okay so let's go through it first thing we check for every time is always the Divergence test so on number one problem number one what's the first thing you you noticing up here the limit's going to equal 2/3 not equals z diverges it's that fast don't worry about looking at telescoping or anything else it's going to diverge does that make sense to you next one do you know you can separate separate series by addition or subtraction then in this case you'd have something that is well that's a geometric this is geometric with two 1/3 to the N do you see it yes this you need partial fractions that's going to be telescoping so you get geometric and you have telescoping both of them would have to converge in order for the series to converge you with me okay so this is geometric and telescoping next one well we go okay geometric no telescoping no but this can look like a p series make them into a type of series if you can Divergence test cool geometric no no problem can make it geometric telescoping partial fractions for telescoping here we have one to the okay this is a p Series where p is greater than one that thing is going to converge it can be that fast next one for something like this well it's really easy to see that this is always positive it's always continuous for n greater than three as it says right here it's always decreasing we don't even have a numerator besides one and the integral would be really e easy because if you use a substitution substitution of = Ln of n gives you 1 n it's right there this is going to be U2 and then we do an integral of that so integral test for this one next up that one well here's the difference between these two what you should do and when you should do them a lot of times when you have single you see what I'm saying about single terms there's not multiple terms here when you have single terms do a comparison test this will be less than < TK n over N2 do a comparison test with that this would be n to the three Hales this is a p series p is uh greater than one it's going to converge therefore because it's always less than that by the comparison test this series also converges does that make sense to you so comparison test here when you have more than one term it's really hard to do this because you don't know what's less than you don't know what's greater than because you have these other terms being added on or subtracted that's really hard does that make sense so with single terms maybe try just the comparison test because it's way easier okay with multiple terms you're going to look at what it looks like how it behaves how it behaves is with the leading terms so we base it on this one then we take a limit of A subn over B subn and we do the limit comparison test we okay with that one are you are you guys sure am I going too fast for you okay next up this is just straight up automatically what Alterna series just do Alterna series t with that it's really easy to see that it is decreasing it's really easy to see that the limit is going to be zero alternating series test and you're done with this one you actually have an option you can do either the root test or the ratio test with this it doesn't matter so pick either one of this the root test is a little tricky because you have to make n be something to the N power power it's possible it's just not easy to do well it's not super hard to do uh you can do this n the 1 is the same thing as n the N / n which is the same thing as n the 1 n to the N it looks weird but you can do that and that way you'd have this this ratio appropriately uh the last one when you get down to something like this please do not start doing weird things like a ratio test or root test when you have a sign in there already is that going to work out for you no it's not alternating so when you have something with a sign or a cosine or something like this do the absolute value first because as soon as you do the absolute value so use absolute value as soon as you do that then you can say well the absolute value of sign is always less than or equal to one does that make sense and then you have this comparison test idea so try absolute value and then you'll be able to use one of these other test I'm put then dot dot dot use something else use a comparison test um did this help you at all this summary show P if you feel okay with that one okay so this will hopefully give you an idea about what you can do uh this is every test that we've done there's a lot of tests but now you have at least the an outline did this section make sense for you all right