Current time:0:00Total duration:4:53
Divergent telescoping series
Lets say that we have the sum one minus one plus one minus one plus one and just keeps going on and on and on like that forever and we can write that with sigma notation. This would be the sum from n is equal to one lower case n equals one to infinity. We have an infinite number of terms here but see this first one, we want it to be a positive one, and then we want to keep switching terms. So we could say that this is negative 1 to the lowercase n minus 1 power. Let's just verify that that works. When n is equal to 1, it's negative 1 to the 0 power, which is that. When n is equal to 2, it's 2 minus 1; it's negative one to the first power that's equal to that right over there. So this is a way of writing this series. Now what I wanna think about is does this series converge to an actual finite value? Or, and this is another way of saying it, what is the sum? Is there a finite sum that is equal to this right over here, or does this series diverge? And the way that we can think about that is by thinking about its partial sums let me write that down. The partial, the partial sums of this series. And the way we can define the partial sums, so we'll give an index here. So capital N, so the partial sum is going to be the sum from n equals one but not infinity but to capital N of negative 1 to the n minus 1. So just to be clear, what this means, so the, the partial sum with just one term is just gonna be from lowercase n equals 1 to uppercase N equals 1. So it's just going to be this first term right over here. It's just going to be 1. The S sub two, s sub two is going to be equal to one minus one. It's gonna be the sum of the first two terms. S sub three, S sub three is going to be one minus one plus one. It's the sum of the first three terms, which is of course equal to. Equal to, let's see this equal to one. This one over here is equal to zero. S sub 4, we could keep going, S sub four is going to be one minus one plus one minus one which is equal to zero again. So once again, the question is, does this sum converge to some finite value? And I encourage you to pause this video and think about it, given what we see about the partial sums right over here. So in order for a series to converge, that means that the limit, an infinite series to converge, that means that the limit, the limit, so if you're a convergence, convergence is the same thing, is the same thing as saying that the limit as capital, the limit as capital N approaches infinity of our partial sums is equal to some finite. Let me just write like this, is equal to some Finite, so Finite Value. So, what is this limit going to be? Well, let's see if we can write this. So, this is going to be, let's see s sub n, if we want to write this in general terms. We already see if s, if capital N is odd, it's equal to 1. If capital N is even, it's equal to 0. So, we can write, lets write this down so s sub n I could write it like this is going to be one if n odd it's equal to zero if n even. So what's the limit as s sub n approaches infinity so what's the limit. What's the limit, as N approaches infinity of S sub N. Well, this limit doesn't exist. It keeps oscillating between these points. You give me, you, you go one more, it goes from 1 to 0. You give me one more, it goes from 0 to 1. So it actually is not approaching a finite value. So this right over here does not exist. It's tempting, because it's bounded. It's only, it keeps oscillating between 1 and 0. But it does not go to one particular value as n approaches infinity. So here we would say that our series s diverges. Our series S diverges.