at last we have reached this last video of the AP Calculus review series specifically for BC students for ab it was Unit 8 but this is Unit 10 infinite sequences and series and it's going to be the last unit of calculus and so uh with this video You're basically done with your Calculus journey and the only thing left is to wish you good luck for your exam uh and we're not done yet though because we left this unit to do and it's a relatively long one not as long as the application of integration or the integration unit itself but this one's up there so with that in mind let's just get started with the content review uh and so starting title uh it says infinite sequences and series what is an infinite sequence or series sequence is just a set of terms and so an infinite sequence is an infinite set of terms we're mainly talking about infinite series though and uh that is a sum of infinite terms that follows a certain path P um for example 1 plus a half plus a 4th plus an eth do you see the pattern here and so on all the way to Infinity that's why it's called an infinite Series this uh is actually a pretty famous example you should know how to calculate this from algebra and pre pre-calculus but uh we will talk about how to calculate this in a minute first there are three types of series first uh a convergence Series has a limit um has a finite limit of partial sums if you take the limit of the partial sums which is just this is the first partial sum second third and fourth if the limit as n approaches Infinity of this partial sum is finite basically it exists then it's convergent if it doesn't exist it's Divergent um basically the limit of partial sums is infinite uh and then the third type doesn't really have to do any with any of these but an alternating series is when the terms alternate between positive and negative you can have 1 minus a half plus a 4th minus an eigh and so on you're alternating between positive and negative and that's an alternating series and so you might know all this from algebra pre-calculus May I'm doubtful you know what convergent and Divergent series or at least the formal definition of them are but you should it's kind of intuitive here but what you don't know definitely are the tests that you use for these different series um there's a bunch of them a handful of them uh and now when you're given a series usually in summation notation like uh this you can find out whether it's convergent or Divergent using a bunch of tests and you're going to just use one of these tests on each series but the hard part is deciding what test to use and then after that is remembering what the different tests are and then um actually using the test and so let's run through all the tests one by one first is the nth term test uh and that basically says that if the limit if the limit as n approaches in Infinity of the sequence itself of a of n is not equal to zero then it diverges uh this is one of the only tests that actually say it diverges as a conclusion most of them say it converges as a conclusion but this one it says it diverges um but that doesn't mean that if the limit as n approaches Infinity is zero that means it converges it's possible that it still diverges if the limit is equal to zero but if it's not equal to zero then you know for sure that it diverges so that's the nth term test also known as the Divergence test and just to conserve space I'm going to be erasing the test after done I'm done explaining about it I'm just going to keep the list of uh tests up on the screen then there's geometric series uh this that I started talking up about earlier is actually a geometric series a geometric Series has a common ratio R um and if this R is less than one then it converges this uh you say it converges if it's less than one uh and so an R of um an a negative R that's less than negative 1 also converges so the absolute value of R has to be less than one uh the third one is alternating series um and we did talk about alternating series just a few minutes ago um if the alternating series has a limit as n approaches Infinity of the absolute value of a ofn absolute value that means you don't care if it alternates anymore is just the magnitude of the term if that is equal to zero then you know for sure it converges it's similar to the end term but more powerful because if it's alternating and uh the limit as the term approaches zero then you know for sure it converges it kind of looks like that it's converging to a certain value okay um well what about the next test uh P series P series basically says if you have a um a series in the form of 1 over uh X to the^ of P for example or n the^ of P or whatever uh probably n the^ of p is better cuz we are going to use x um for something else uh n is usually the variable we use um this converges if p is greater than one if it's less than or equal to one it diverges and so now we have enough information to answer this uh this one has n the^ of 1 and so p is less than or equal to one which means it diverges um it still approaches zero uh limit as n approaches Infinity of 1 n is equal to Z which means which actually kind of proves what I was saying about the nth term test because just because it approaches zero doesn't mean it uh it is convergent but if it doesn't approach zero then you know for sure it's not convergent so yeah you can use the P series for something like that okay fifth test uh this is just the first four out of like eight of them uh this one's called the direct comparison test and basically what it is is you have a known series you know if it's converges or diverges and you have this unknown series that you want to see if it converges or diverges if this unknown series is less than or equal to the known series and the known series converges then the unknown series converges similarly if this unknown series is greater than or equal to this known series and the known series diverges then the unknown series is also diverges another way you can say if a series is less than a converging series or greater than a diverging series then it has the same convergence as that other Series this is pretty useful because if you know that 1 / n diverges for example and 1 / n minus one is always going to be greater than or equal to 1/ n because you can plug in a few terms and see that 1/3 is greater than equal to 1/4 for example that means 1/ n minus n minus 1 also diverges okay but what if you have 1 / n + one well that's less than or equal to 1/ n but one over n diverges so we cannot use the direct comparison so we have to use its cousin the limit comparison test um and the only reason I'm calling it its cousin and not its brother for example uh is because it's it has the same name but it's not that similar all it does is it Compares two series but it uses a different method direct comparison is just inequalities limit comparison you take the limit as the name suggests if the limit as n approaches Infinity of the quotient or ratio of these two terms um is finite but not zero so if it's zero or Infinity then it's inconclusive but if it's finite then it has the same convergence as the known Series so let's try that with this example over here um that I was just talking about here we can use the limit comparison test so you do limit as n approaches Infinity of 1 / n over 1/ n + 1 and so if you were to simplify that that's just limit as n approaches Infinity of well this is just n + 1/ n CU you're swapping the denominators when you're um simplifying the complex fractions and so this is equal to one and that is finite but not zero and so it has the same convergence the original series diverges and so this also diverges and so you can use limit or direct comparison test when they're pretty similar uh but sometimes direct comparison doesn't work note that if it's 1/ - 1 you can also use limit comparison you just cannot use direct for 1/ n + 1 limit is just a bit more powerful than direct I would say uh and then the seventh test the integral test here we're getting back into our calculus Roots we're talking about integrals we have actually diverged um no pun intended from calculus a bit we've been talking about how to find if series or convergent or Divergence we have talked about limits and things like that but no we haven't talked about derivatives or integrals in a while and finally here we are integ growth um it says that if it's a continuous positive decreasing series all three of those conditions have to be met and usually they will be but always make sure to check continuous positive decreasing um and you have uh the summation of that series AF of n then it has the same convergence as the improper integral with the same bounds so let's say this is k equals 0 to Infinity uh right or n equals 0 to Infinity it's going to be 0 to Infinity of f of n that has the same exact thing so if this is 1 / n for example this would be 1 / X DX uh would be the improper integral you calculate that if it converges then this also converges if that diverges then the series also diverges now notice if this converges to a specific value like one it doesn't obviously this is just an example that doesn't mean that the series itself converges to one it just means that it converges important distinction to make over there uh and then finally most powerful test I would say um is the ratio test uh you're going to use it a lot in the second part of this unit which is about power series and things like that it says um that sorry um yeah with power series uh this is called a ratio test sorry about that um if limit as n approaches Infinity of the next term over the current term is less than one than a converges if it's equal to one it's inconclusive if it's greater than one it diverges um and so you're going to need to kind of use this a lot when we're getting into the next topic but um usually you use this when you have a bunch of NS like in n in the exponent n factorial things like that uh it's easy to cancel off with the next term so those are the eight tests you should know um know how to use know when to use and things like that okay uh one more thing about this absolute convergence is when um if the series absolute value of a of n converges then the series a of n also converges you know this for a fact if the absolute value of afn converges then just afn converges as well and so if afn is an alternating series you can take the absolute value of it see if that converges and if so then this converges as well you can prove that easily with the N term test by the way uh and so in light of this you can describe a series in three different way first converges absolutely which means that both of these converge converges conditionally which means that only the alternating series converges but the absolute value um absolute value does not converge uh and then diverges which means both of them diverge and I can give a few examples of both of them of all three of them if you have -1 ^ of n/ n^2 this is Absol absolutely converges cuz this converges cuz when you use the alternating series test it approaches zero which means it automatically converges if you take the absolute value of it you get 1 over n^2 which by the P series test also converges but if you were to remove this two here you would get 1^ n / n which does converge by the nend term by the alternating series test because the limit approach is zero but when you take the absolute value of it you get one over n which we already proved diverges because of the P series test and then you can find a bunch of examples of um uh series that diverge uh just without conditionally converging for example1 the^ n just times n you automatically know this is going to diverge because it doesn't even satisfy the alternating series test where the limit as n approaches uh infinity of this approaches Infinity cuz there's this n here instead of in the denominator so good thing to know absolute convergence okay finally the second part of this unit we talked about tests and how to use them now we're going to talk about power series tailor polinomial mearn polinomial error bounds fun stuff like that we're getting back into calculus uh not yet actually power series is not strictly calculus but um it's basically when you have X in there you have uh some series afn and then you have this x - c to^ of n term um and so an example would be uh x - 2 ^ n Over N factorial this is a power Series where one over n factorial is this all other series a of n that you're multiplying by and here x - 2 to the^ of n is x - c to the^ of N and you can expand a few terms do some stuff with this uh but what we're mainly going to do is um find the interval and radius of convergence so the interval of convergence is basically the interval for which power series converge and the radius is half of that interval and usually when you're determining the um interval of convergence you use the ratio test uh because most of it most of the um examples which is going to be like the series of x - 2^ n/ n factorial it has an N factorial ratio test is pretty useful here you can do next term minus Uh current over current term limit of that uh and then see what that's equal to and usually it's going to be less than one or no you set it less than one and solve for x when it's uh like that uh and so quick example of that if you were to use the ratio test limit as n approaches Infinity of this I'm just doing an example because it's a pretty important topic um you do n + 1 over n +1 factorial and then since you're dividing by a fraction you're multiplying by its reciprocal which is just n factorial over x - 2^ n uh and things cancel out you get limit as n approaches Infinity of x - 2 cuz that cancel out with that over n n + 1 and you want this to be less than one because you're trying to find out where it converges and so uh this just 1/ n + 1 always is less than one and so you want um uh this is just limit as n approaches Infinity of x - 2 * 1 n + 1 this approaches uh zero and so you have absolute value of xus 2 um just by itself uh is less than one uh and actually in this case uh it's not going to be as less than one this just converges for all values because um this approach is zero which means this entire expression approach is zero Which is less than one so this is a bad example for a power series be honest but if it was something like n / n + 1 you somehow got that as the limit you would do this approach is one and so all that's left is this absolute value of x + 2 Which is less than one and so now you solve this inequality uh for X it's an absolute value of inequality which you should know to how to solve from algebra and so1 is less than x - 2 is less than 1 and so uh you can solve for x by adding two to both sides you get 1 is less than x is less than three but we are not done the interval is 1 to 3 but what is a closed or open um and so you test the end points by plugging in one into this if you plug in one into this you get 1 - 2 is1 to the^ of N and so you have uh series1 to the^ of n Over N factorial does that converge yes it does by alternating series test and so this is closed here um and obviously uh again I made a edit to here so we're not plugging it into the actual series that gave us this um interval this is just an example I should have chosen a better example in hindsight but um this is how you would do it you would solve for the interval and then you would plug in the end points back into the original series uh to see if x equals this certain value will the entire series converge or not so that's just how to use power Series in a nutshell uh but we can apply power series to a much more uh interesting topic and that is pols uh we're going to approximate polinomial using what are called Taylor and mclen series now ding ding ding remember I told told you way back uh when probably in unit 4 um about how a bunch of calculus was about approximating things approximating values of functions using tangent lines then using Oilers method now we're going to be talking about tailor pols to approximate functions this is going to be the best um method we've done so far which is why we're learning it last and so how do you construct a tailor polinomial is just using the nth derivative of a certain function um so f ofx is equal to the series uh of and this is going to get complicated pretty fast so it's the nth derivative at a point a Over N factorial * x - A to the^ n and this is a power Series this is no matter how weird it looks it's all in terms of N and then there's this x term meaning it is a power series uh a here is the point you're centering it around um this is a tailor polinomial with infinite terms but if you wanted a finite one without using this you would just use um the nth derivative of function function so if you have X Cub + 3 this is actually a bad example cuz that is a polinomial itself let's say you have S of X right um You would construct the tailor polinomial by saying well what is sign of let's say we're centering it around one or Pi Pi is easier s of Pi well that's just going to be equal to zero uh and so this is zero plus uh and then you plug in what's the derivative of s that's going to be cosine cosine of pi is ne1 and so you do -1 * x - piun to the^ of 1 over 1 factorial uh and so it's basically going to be this exact formula you just are doing it for a finite polinomial instead of an infinite series uh and so you can find like the first three terms by just plugging in the third derivative or yeah third derivative of the function and then xus a ^ 3 over 3 factorial and so on so that's how you would calculate the tailor polinomial mcloren pols are a specific type of tailor pols that are Center at x equal Z and they're much easier to solve for U that's going to be equal to uh the series of FN the N derivative add zero Over N factorial times just X the^ of n because a is zero in this case uh and so that's going to be your muren series and so I'm kind of speeding through this cuz uh this is just a Content review uh and it's already going pretty long uh longer than I expected and so these are the two types of series that you should know how to construct from a given function but there are also a few you should memorize uh if it's e to the X that's equal to the series X the^ of n/ n factorial uh if it is cosine of x that is equal to the this is um a bit more complicated 1^ n it is alternating * x^ 2 n/ 2N factorial this is cosine s uh that's going to be pretty similar it's going to be 1^ n * x^ 2 n + 1/ 2 n + 1 factorial uh and then we have 1 over 1 - x which is just the series of just X the perf and very simple and so you might be wondering why do we need this weird function well that's cuz it just gives us a really nice mear series uh and that's not all using these four you can um use substitution differentiation and integration to calculate others for example how do you find inverse tangent of X well you know that um inverse tangent of X plus C technically uh is equal to the anti-derivative of 1 over um 1 + x^2 for example DX and so uh what do you do there well using this you can plug in X is equal to x^2 that looks really weird from a mathematical standpoint but we're just plugging an X into here so 1 over 1-x^2 which is just equal to 1 + x^2 which is what we want so we can find the Tailor's um or the muren series for 1 over 1 + x^2 by just plugging in negative x^2 wherever we see an X in here uh and then we take the anti-derivative of that resulting polinomial and we're going to get the uh Taylor or mcen series for inverse tangent of X and finally aor bounds so you have all these nice formulas how do you find out how accurate they are um and so there are two types uh one for alternating series and one for just General functions so you can use um alternating series with cosine and S uh and you would use lrange for example with e to the power of X I also erased 1 over 1 - x cuz the only purpose that served was for the inverse tangent der derivation you still should memorize it but it's not as important as the others it's also easy to it's kind of easy to derive okay uh so yeah alternating series Airbound so if you have a function f the actual value of f of C minus the tailor polom at C the absolute value of that that's just going to be the error the maximum value of that error is going to be equal to the absolute value of the n + first derivative at C Over N +1 factorial * c - A to ^ n + one and so a is where you're centering the tailor series around uh or tailor polinomial tailor series are always exactly accurate tailor polinomial are the finite representations of Taylor series and so they are not always accurate and so at a certain nth term let's say you're doing a fourth degree tailor polinomial you do the fifth derivative uh and then you do five factorial down here to the power of five here and so on um that's for alternating this is just going to be the next term of the uh alternating series uh just the absolute value of the next term uh because as you can see this is what you do uh well technically this one for um Center Durant a and so that's just exactly the same thing but you just plug in N plus one instead of uh n and C instead of a because um you're plugging it at a certain point that is not what you're centering there and there's lrange error bound which uh if it's not alternating you use this uh this time it's less than or equal to and it's m/ n +1 factorial * c - a power of n+1 um and that looks very similar but you see this Big M here what is M uh and so it is going to be very similar to this term the N plus first derivative at C but usually um alternating series are kind of easy here because you're just plugging in the next term and so you don't really usually need to calculate this you just find the next term by plugging it into the tailor polinom they'll usually give you an extra term and then ask you let's say they give you a fifth degree polinomial they ask you what's the error for the first four terms you just plug it into the fifth so you don't really need to do anything but for larange error it's not going to be the next term and so what you're going to need to do is uh find the N plus first derivative which can be pretty hard and so especially since you're going to do this without a calculator most of the time since the entire point of um approximating functions is doing it without a calculator and so um how are you supposed to find the value of the N plus first derivative without a calculator well that's why m is the maximum value of the N plus first derivative on a certain interval so let's say it's from 0 to 1 it could be one but let's say it's something like e right you don't know the exact value of e the^ 1 you might obviously maybe it's like e to the^ of 0.5 for example you don't know the exact value of that but you can say e to the power of 1 is less than three definitely cuz this to be honest is 2.7 approximately 2.718 but you know it's going to be less than three and so you say m is equal to 3 and then plug that into so you're just making a conservative estimate for this error uh it's usually going to be less than that but sometimes you have the exact value of the function and you can just plug that into okay uh and yeah that is it for a Content review it took way longer than I expected uh but that is a pretty thorough explanation of Unit 10 it's time for our practice problems we have four this time uh the first two are both testing a series for convergence uh which is a pretty big chunk of this unit to be honest so let's start with the first one um we could use ratio test for example let's try that out limit as n approaches Infinity of the next term which is n + 1 over technically n + 2 to^ n + 1 because n + 1 + 1 is n + 2 * reciprocal which is n + 1 ^ of n over n and that's pretty hard to evaluate but what we do see is an N the^ of n * a n which means it's approximately n the^ of n + 1 there are other terms that follow it but all we care about is the leading term when we're talking about limit as n approaches Infinity here we have n the^ n + 1 * an N which gives us um n the^ n + 2 which simplifies down to 1/ n and so you can say that this limit is equal to zero because uh of this it's kind of hard to show but it is valid um you if you can somehow explain on the AP test you can use this or you can try using other tests as well and so this is less than one and so this is a convergent um ratio a convergent series based on the ratio test okay uh this second one okay uh technically we can say that this first one converges absolutely because even though this is not an alternating series absolute convergence is not limited to alternating series it just we usually um associate them with it because we see an absolute value there and so technically the absolute value of this is it itself and so both of them do converge meaning it does converge absolutely we can say that here here it also uh it makes even more sense to use absolute convergence so we will try that we will see um if it converges absolutely conditionally or diverges for this this one um we can use the alternating series test limit as n approaches Infinity of absolute value of this which cancels out the alternator so n + 1 / 7 n^2 - 5 uh we see an n and an n^2 here which means this does equal zero and so it does converge so we see this converges but does it converge absolutely let's take the absolute value of it so in uh series of n + 1/ 7 n^2 - 5 does this converge or not um one way to figure it out is limit comparison test because we see an n and n^ S so it's like having a 1 / n um which we know diverges but let's see if this is similar to that using the limit comparison test and so limit as n approaches Infinity of n + 1/ 7 n^2 - 5 and so we're dividing by a fraction meaning multiplying by reciprocal of n / 1 which is equal to limit as n approaches Infinity of n^2 + n / 7 n^2 - 5 and so uh you know that's going to be equal to 17 cuz n^2 7 n^2 is the ratio of the first terms which is a finite um which is a finite number uh that is not equal to zero and so it does have the same convergence as 1/ n and so this one diverges meaning this converges conditionally only okay what about this next one uh it doesn't give us any instructions but the instructions for this problem should be find the mclen series of this uh we actually kind kind of did this I didn't show you what it should be but we know how to do it and so we know 1 1 - x is equal to uh this right which you can also write as 1 + x + X2 + x Cub plus dot dot dot and so 1 over 1 + x^2 is going to be using substitution you say x is equal to x^2 and so every time you see an X here you're going to replace it with x^2 to the^ of N and so it's going to be uh -1 to ^ of n n * x ^ of 2N and so this is going to become an alternating series now so that's going to be e approximately uh 1 - um x^2 + x ^ 4 - x ^ 6 Plus and so on so that's going to be our Meen series for uh this function okay and this last one has to do with error uh this is an alternating series which means we can use the alternating series airror bound um it says it's approximated by S of 50 so we use the 51st term uh we can just directly plug in um s 51 into this cuz the sum uh this technically is not a um a mclen series or a Taylor series or um sorry a tailor polinomial or anything like that it's just the sum of a series and you can still use error for that the sum of a series is just going to be equal to the value of a function um but this one isn't explicitly about a function you can still use error for that usually it's not common but it is um possible and so the sum of the first 50 terms of this series is approximately equal to the entire term uh and the error of that is going to be the the maximum error of that is going to be the 51st term and so we just plug in 51 into here1 to ^ 51 - 1 over 2 * 51 - 1 so1 ^ 50 over 102 - 1 which is 101 so that uh cancel out that and it's also going to be the absolute value and so even if it wasn't negative one to the numerator it would be um the absolute value of that which is just one and so 1 over 101 is going to be the maximum error that doesn't say is the error it's going to be the maximum error the error is always going to be less than this value and unfortunately that is it for AP Calculus all of the units uh this is the last video like I said in the beginning uh but yeah and so good luck on your AP uh test hopefully all this practice helped you out but always seek more practice do more worksheets uh learn more content this is just just uh to refresh your minds and give you a taste of what is coming on the AP exam so yeah good luck on the AP exam and I'll probably see you in another video