 Welcome back to our lecture series math 1220 calculus 2 for students at Southern Utah University as usual I'm your professor today. Dr. Andrew Misseldine this lecture represents the first one for lecture 37 in our series Now this one is going to be a continuation of lecture 36, which we're talking about Series that is an infinite sum of a sequence in lecture 36. We talked a lot about geometric series So we're going to change pace a little bit In this video, we're going to talk about the idea of a telescoping series and what we can do with them So consider the series the sum or k ranges from 1 to infinity of the sequence 1 over k times k plus 1 Now when it comes to series it's very important to see similarities between series and integrals because Series in essence are just a different type of integral. It's not a continuous integral It's now a discrete one, but a lot of the techniques that we used when we talked about integrals actually apply to series as well So for example, if we had the improper integral from 1 to infinity of x of 1 over x times x plus 1 and we want to integrate this thing Our our thought would probably eventually get to the idea that oh we should do some type of partial fraction decomposition in order to Calculate the antiderivative it turns out finding the convergence of the series is going to require the same thing So if we take the fraction 1 over k times k plus 1 and we wanted to do a partial fraction decomposition We'd get a over k plus b over k plus 1 as our template clearing the denominators we get 1 equals b times Sorry, we'll start with a the first letter of the alphabet there a times k plus 1 And then add to that b times k. We're going to pick some cool numbers to annihilate with for example If we take k to be zero that annihilates the bk Leaving only McDonald's is the option and that is to say we get one One is equal to just a right and then the other option if we take a or k to be negative one Negative one that'll annihilate the a leaving only behind b in which case we get that negative b equals one That is b equals negative one so with that in mind our partial fractions look like the following like We saw a moment ago a is equal to one and b is equal to negative one I'm actually just going to put the negative sign here in the middle Like so and so this gives us our partial fractions now if this was an integral We would break it up into integrals and continue like that So we're going to try a similar thing for this one our series Our series where we take the sum from k equals 1 to infinity of 1 over k times k plus 1 This actually breaks up into two series. We have the series. Well, we have the series where k goes from 1 to infinity of 1 over k, but then we subtract from that the series k equals 1 to infinity of 1 over k plus 1 Now let's look at what happens to this thing as we expand it we expand it out So if we look at the first term of this series, we would end up with something like well when you plug in k equals 1 You're gonna get 1 minus You're gonna get 1 over 1 plus 1. That's a 2 So we'll just simplify that so you get a 2 right here. And so this is our first term That is k equals 1 in that situation So maybe I'll be specific about that k equals 1 Then if we look at the second term, we're gonna end up with 1 half. That is 1 over 2 And then we're gonna subtract from that 1 third Where the 1 third comes from plugging a 2 into this expression 1 over 2 plus 1 we get a 1 third This is now our second term k equals 2 and if we do a few more just to get the feel of what's going on here We're gonna end up with 1 third minus 1 fourth That is our third term right here k equals 3 our next term would look like 1 fourth minus 1 fifth And then this pattern is going to continue on and on and on Add infinitum So this is our fourth term again. There's there's more and more of these things Now an interesting pattern occurs when we look at this series and this is actually where we get its name What's going to happen? You'll notice that in the first group you have a negative 1 half But in the second group, there's a positive 1 half when we add those things together They're actually going to cancel each other out When we look at the the second group and the third group The second group has a negative 1 third The third group has a positive 1 third when you add those together They're going to cancel out as well Looking at the third and fourth group They both have a negative one has a negative one fourth and the other has a positive one fourth When you add those together they likewise would cancel out and although I didn't write it on the screen If we compared the fourth group with the fifth group Both of those will have a one fifth one is negative like you see here The next one would be a would be positive They're going to cancel each other out and you see this happening over and over and over again And so if we were to just consider a partial sum if we just take s in to be the partial sum We're going to take the sum where k equals 1 to n And we take these two sequences 1 over k minus 1 over k plus 1 This would end up with the same basic idea you'd end up with the first term The one that we see floating around right here But then eventually this thing would stop with the very last term if we went to the nth term We're going to get this 1 over n plus 1 because this pattern again would keep on going cascading cascading down Until the very last term here wouldn't cancel out if we had this partial sum And so because this is the partial sum of our series If we take the limit here as n goes to infinity This thing will look like 1 minus 1 over infinity for which that part is just 0 Right the second piece just goes to 0 and we end up with this series Adding up to the number 1 And so this shows us that the series is in fact convergent because the limit of partial sums exist Because remember that's what that's what a series is the series itself is the limit of the partial sums And so if the limit of the partial sums Exists that's what the series is and so we say the series is convergent when this limit exists All right, and so we're going to get that that limit is equal to 1 Now this example demonstrates to us the idea of a telescoping series That's what it's often called here telescoping summer telescoping series And the idea comes from the following metaphor If we think of like a pirate sailing the high seas and you have A crewman in the crow crow's nest looking upon the horizon Your typical spyglass looks much like the following type object here, right where you have a lens Here on the outside and then you have a small little lens right here for which the crewman would look through Now when it's completely expanded you see all these multiple chambers But then when you're done with the spyglass You're going to collapse it down in which case you don't see Really any of this anymore these parts Just kind of vanish away and everything just collapses inside of this one chamber So you still have the lens on the left you have the lens on the right But then everything else seems like it vanishes away That's that that's that's the metaphor we're using for this telescoping series right here That you have this you have this first term right here And then there's going to be some final term But then everything else cancels out when you collapse it down at the moment This thing looks like the expanded spyglass Right here, but then this thing right here Looks just like the collapse spy get spyglass We just have this first term and this last term And the nice thing about a telescoping series is that if you look at the partial sums you're going to get a You're going to get a general form For the partial sums for which we can then very easily take the limit as it goes to infinity And oftentimes not always oftentimes the tail end is going to vanish vanish away Now one has to be very careful with this because the series itself is a limit Of the partial sums but when you look at the original form It can be very tempted to be like oh, yeah, here you go It starts off with one everything else is going to cancel out so it's going to equal one You would be right in this situation, but sort of somewhat of a misleading perspective As an example to compare remember you take the series Where k equals one to what we'll start at zero from zero to infinity And you take negative one to the n And expanded form this thing looks like one minus one plus one minus one plus one minus one plus Yeah, et cetera, et cetera, right? And so this one can be very tempting to think that oh, it's a telescoping series, right? The one and negative one cancel the one and negative one cancel the one and negative one cancel This thing should add up to be zero But we looked at this example before right you could take a slightly different perspective and be like Oh the negative one and one cancel the negative one and one cancel the negative one and the one cancel And so this thing is going to add up to be one it clearly can't be both And that's because as we saw before this series is divergent But if if you're not careful With these telescoping series you can fall into the same type of trap we have right here So the the strategy that I will encourage you to employ is come up with a come up with the partial sums Don't go off towards infinity. That's when things can get kooky Go to a finite value if we terminate at the nth step Then everything will will cancel out except for the one and the last term the negative one over n plus one And then with this partial sum formula in hand then take the limit And you'll see with that perspective you won't get the wrong Convergence because what happens with you have a divergent telescoping series You might be come to believe that it's convergent when in all reality it's divergent. So watch out for that