**4.9 Properties of Convergent Series — Transcript & Summary | SozAI**
Source: https://sozai.app/transcript/properties-convergent-series-2/

Explore key properties of convergent series including linearity, tail convergence, and important theorems with proofs and exercises.

## Key Takeaways

- Linearity of convergent series allows addition and scalar multiplication while preserving convergence.
- Convergence is unaffected by removing a finite number of initial terms.
- The behavior of tails of series is crucial to understanding convergence.
- Proofs rely on properties of finite sums and limits of partial sums.
- Understanding these properties aids in mastering series convergence concepts.

## What the video covers

- Convergent series are linear: the sum of two convergent series is convergent and equals the sum of their sums.
- Multiplying a convergent series by a real constant results in another convergent series with the sum scaled accordingly.
- Proof of linearity property is demonstrated using limits of partial sums and finite sum properties.
- Only finite sums can be split using commutativity and associativity before taking limits.
- If a series converges, then its tail (series starting from any finite index) also converges.
- Ignoring a finite number of terms does not affect the convergence of a series, though it changes the sum.
- A series converges if and only if the limit of its tails goes to zero.
- The video encourages viewers to attempt proofs themselves to deepen understanding.
- Definitions of series as limits of partial sums are emphasized for understanding convergence.
- The distinction between finite sums manipulation and infinite limit processes is highlighted.

## Chapters

1. 00:00 Introduction to Properties of Convergent Series
2. 00:30 Linearity of Convergent Series: Sum of Two Series
3. 02:00 Proof of Linearity Using Partial Sums and Limits
4. 03:30 Scalar Multiplication of Convergent Series (Exercise)
5. 04:00 Convergence of the Tail of a Series
6. 05:00 Theorem: Series Converges iff Tail Limit Goes to Zero
7. 06:00 Encouragement to Attempt Proofs and Summary

Answers

## Questions about this video

What does it mean that convergent series are linear?

It means that the sum of two convergent series is also convergent, and the sum of the combined series equals the sum of each series added together. Similarly, multiplying a convergent series by a real number results in another convergent series with the sum scaled by that number.

Does removing a finite number of terms affect the convergence of a series?

No, ignoring a finite number of terms does not affect whether a series converges or diverges. It will change the actual sum, but the property of convergence remains unchanged.

What is the significance of the tails of a series in convergence?

The tails of a series refer to the series starting from some finite index onward. A key property is that a series converges if and only if the limit of its tails goes to zero, meaning the partial sums of the tail get arbitrarily close to zero as you move further along the series.

## Full Transcript — Download SRT & Markdown

00:01

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.

00:18

Speaker A

convergence series some of the a n's and some of the ends 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

00:30

Speaker A

the series which is very nice but i repeat here it is important that we're assuming that this both series are convergent if this is not the case we cannot say it similar to what happened with limits remember and then we

00:47

Speaker A

also say that the series of c times a n is convergent and that the sum is c times the sum of the a ends so i'm going to prove the first one and i will leave the second was an exercise

01:02

Speaker A

you see we know that the series of the ans 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 pn's converges so the sequence of the

01:17

Speaker A

partial sums converges and let's call it like tk'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 showing that the limit of those partial

01:36

Speaker A

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

01:54

Speaker A

allowed there's no problems 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 of the sum of the limits because

02:10

Speaker A

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 aliens and the series of the pn and that completes the proof as you see

02:23

Speaker A

it is a quite 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

02:36

Speaker A

that the sum from n equals 1 to k of a n plus b n we could split it at the sum of the amps by the sum of the bm's so that we were using commutativity and associativity quite heavily but that is

02:49

Speaker A

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

03:05

Speaker A

important property is that we know that at series converges even only like the tail converges so what it says is like for example if m is eight like summing a1 a2 plus a3 plus a4 forever it's pretty much if that converges then

03:26

Speaker A

something like seven a8 plus eight nine 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 conversion and

03:44

Speaker A

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 is convergent or divergent is not going to change and that's actually going to be very

03:57

Speaker A

useful and there is another theorem that is quite nice says that series is convergence if and only if the limit of these tails goes to zero what means is like if we throw away many many terms like it gets

04:14

Speaker A

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

04:30

Speaker A

try to write down what it means for these tails to go to zero and try to put all the pieces together

Topics: convergent series series linearity partial sums series tails series convergence proof mathematics Beatriz Navarro Lameda infinite series limits math tutorial

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