Original subtitles

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Today we're going to be talking about how to determine whether or not the telescoping series is convergent

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or divergent.

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And if it is convergent how to find it some.

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And in this particular problem we've been given this telescoping series which is the infinite sum from

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unequals 1 to infinity of our series here which is either the one or end power minus either the one

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over and plus one power.

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Now the best way to tell whether or not you actually have a telescoping series on your hands in the

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first place is to plug in the first several terms of the series starting with whatever we've been given

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here which in our case is an equals one.

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So because unequals one we're going to be starting with one and we're just going to plug in the first

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several terms of our series to see what kind of series we end up with.

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So when we plug in one we get either the one over 1 or just eat the one so that's easy.

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When we plug in one to this second term here we get one plus one in our denominator which is two.

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In other words either the one half so we get minus the race to the one half power that's our first term

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in the series.

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Now what do we do we plug in an equal to what we get here for our first term.

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Either one half so easy to the one half minus when we plug into two plus one is three.

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So we get to the third So minus the two the one third.

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Now what happens if we plug in and equals three.

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So Yet here we have a one third minus the two the one fourth and you can start to see a pattern a little

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bit.

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Now it should become obvious fairly quickly that this is a telescoping series because what we see if

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we look at this series is that the second value in the first term is always going to be the negative

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version of the first value in the second term.

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So here we have negative IID of the one half.

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Here we have positive either the one half if we added these all up together they would cancel with one

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another right they would net to zero.

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Here we have negative E to the one third and positive to the one third.

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So those would cancel with each other they'd also net to zero and we could keep going.

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We'd cancel every term in between if we continue on with this series until the end of the series if

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it hadn't and what we would get to for our end term is our original series here right.

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Our last term in the series would be the one over n minus the the one over and plus one that would be

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our last term.

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OK.

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So if we have that what we see is that we're going to end up canceling when you have a telescoping series

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and this is happening and these terms are canceling like this you're always going to end up canceling

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every term in the middle of the series because this one is going to cancel with the first value in the

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next term.

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This either the one minus and value is going to cancel with the second value in the previous term that's

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going to go away as well.

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And all you're ever going to be left with is the very first term and the very last term.

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And so what we have here for the value of the partial sum or the series SLBM is equal to our first term

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here.

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E minus E to the one over and plus one minus E to the one over and plus one.

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Now a few of them were from before we talked about a series of partial sums remember that our original

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series here we call a sub that's the original series to the one over and minus the one over end plus

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one that's our original series a seben when we're talking about the series of partial sums we denoted

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as Subhan.

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And what we're talking about is adding all of these terms together and getting a model for that.

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Notice that this is a different value than our original series.

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A7 this is a representation of the series of partial sums as Subban.

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The cool thing about a telescoping series is that if you can find an explicit value for the end partial

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sum that alone proves that the telescoping series is convergent that's all you need.

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So as long as you can find this and you can easily find a value by cancelling all these terms in the

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center and being left with just the first term and the last term like this and you get a model for this

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series of partial sums you get this as value as long as you find this you know that the telescoping

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series is convergent and that's essentially your convergence test for telescoping series.

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So that's going to be our proof that the series is convergence.

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We can say it's convergent now.

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The only thing we need to know is what the some of the series is.

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Well it's really easy once we have a model for the series of partial sums because if you'll remember

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the infinite sum from any quolls 1 to infinity of our series Sabayon this is just a definition.

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When we have a series A7 and we're looking for the infinite sum here that's going to be equal to the

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limit as NGOs to infinity of the series of partial sums as Subban.

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So this is a convenient sometimes convenient easy way to find the sum of the series if we can find a

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model for sub in the series of partial sums.

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All we have to do is take its limit and approaches infinity and that'll give us the sambar original

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series.

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So we're going to change this year the limit is and goes to infinity of Esben.

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We're going to change that into the limit as and goes to infinity or our actual value for s 7 which

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is B minus the to the one over and plus 1 power.

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So we get that.

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OK so now what happens if we evaluate this limit if we plug in a very very very large value for n where

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when we do that here in this xponent we'll get essentially infinity or a very large number plus one.

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So in other words just still a very large number in this denominator here one or any constant over an

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extremely large number that's always just going to tend to zero.

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This is going to become zero here when that happens.

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We were left with E to the zero power zero power is just one.

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So this becomes our first value here.

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E t minus E to the zero power or in other words the minus one.

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And this is the sum of our series which we just found by taking the limit as and goes to infinity of

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our series of partial sums as 7.

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The value we get for that in our case minus one is the sum of our original series The sum of the series

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a Subhan.