Speaker A
In this video, we're going to study some important properties of convergent series. We'll see that convergent series are linear, and we're also going to study what happens to the tail of a series. So, let's get started. So, we know that we have two convergent series, some of the a_n's and some of the b_n's, and c is a real number. Then, we know that the series of the sum of those terms is also convergent, and like the series of the sum is the sum of the series, which is very nice. But I repeat here, it is important that we're assuming that both series are convergent. If this is not the case, we cannot say it, similar to what happened with limits, remember? And then we also say that the series of c times a_n is convergent, and that the sum is c times the sum of the a_n's. So, I'm going to prove the first one, and I will leave the second as an exercise. You see, we know that the series of the a_n's converges. That means that the limit of the partial sums exists and is equal to, let's say, s. Then, we also know that the series of the b_n's converges, so the sequence of the partial sums converges, and let's call it like t_k's, and the limit is t. Okay, we know these two things. Now, we want to show that this series is convergent, which is the same thing as showing that the limit of those partial sums exists. So, how can we do that? Well, here, the sum from n equals 1 to k of these a_n plus b_n, I can't split them because this is now a finite sum, so all the things that we know about sums are allowed. There's no problem there. Oh, pretty good. And now, we know that the limit of the partial sums for each one of my series exists, so then I can split and say the limit of the sum is the sum of the limits because I know that the sequence of partial sums are both convergent. Oh, but that is just s plus t, but that is indeed the sum of the series of the a_n's and the series of the b_n's, and that completes the proof. As you see, it is quite a simple proof. You just have to be careful with the fact that you are only allowed to do certain things to finite sums, and after you do that to a finite sum, like in this case, we said that the sum from n equals 1 to k of a_n plus b_n, we could split it as the sum of the a_n's plus the sum of the b_n's. So, that we were using commutativity and associativity quite heavily, but that is allowed because that is a finite number adding infinitely many terms. And after we've done that, is that we take the limit, so we push it to infinity, and that is a very important distinction to keep in mind. Another important property is that we know that if a series converges, even only the tail converges. So, what it says is like, for example, if m is eight, like summing a_1, a_2, plus a_3, plus a_4 forever, it's pretty much if that converges, then something like from a_8 plus a_9 and that forever also converges. So, what this is telling me is that ignoring a finite number of terms doesn't affect convergence. It will affect the final sum, so we know it's convergent and the sum would be a number. Of course, if you are throwing away some terms, the actual sum will be different, but the fact that it is convergent or divergent is not going to change, and that's actually going to be very useful. And there is another theorem that is quite nice. It says that a series is convergent if and only if the limit of these tails goes to zero. What it means is like if we throw away many, many terms, it gets closer and closer and closer to zero. I invite you to try to write down the proof of this. It's quite exciting to see it when you get it yourself. Try to write down the definitions of the series as the limit of partial sums. Try to write down what it means for these tails to go to zero, and try to put all the pieces together.
