Calculus Two: Sequences and Series is an introduction to sequences, infinite series, convergence tests, and Taylor series. The course emphasizes not just getting answers, but asking the question "why is this true?"

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From the course by The Ohio State University

Calculus Two: Sequences and Series

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Calculus Two: Sequences and Series is an introduction to sequences, infinite series, convergence tests, and Taylor series. The course emphasizes not just getting answers, but asking the question "why is this true?"

From the lesson

Alternating Series

In this fourth module, we consider absolute and conditional convergence, alternating series and the alternating series test, as well as the limit comparison test. In short, this module considers convergence for series with some negative and some positive terms. Up until now, we had been considering series with nonnegative terms; it is much easier to determine convergence when the terms are nonnegative so in this module, when we consider series with both negative and positive terms, there will definitely be some new complications. In a certain sense, this module is the end of "Does it converge?" In the final two modules, we consider power series and Taylor series. Those last two topics will move us away from questions of mere convergence, so if you have been eager for new material, stay tuned!

- Jim Fowler, PhDProfessor

Mathematics

The comparison test, again.

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Well, here's something that happens.

Well, maybe you're analyzing a couple series, and

the first thing that you do is try to

apply the limit test, and you find that both

cases, the limit of the nth term is zero.

So, at least these series aren't diverging for an obvious

reason, like the limit of the nth term is non zero.

So you got two series that may or may not converge.

But, even though the limit

of a sub n is zero, and the limit of b sub n is zero, it could be that the limit as n

goes to infinity of a sub n over b sub n might be some number L, which is positive.

What does that really mean?

Well one way to think about that is that it's sort of saying something like this.

It's saying that a sub n, b sub n are almost multiples of

each other, like a sub n is almost a multiple of b sub n,

at least when n is really big. I can be more precise with epsilons.

So let's set epsilon equal to L in this case.

Epsilon's going to be a positive number, but

I'm assuming that my limit, L, is positive.

So let's set epsilon equal to L. Then the definition of limit says what?

It says that there's some big N, so that whenever little

n is greater than or equal to big n the distance

between the thing I'm taking the limit of, a sub n

over b sub n, and my limit, L, is less than epsilon.

And in this case, right, epsilon is L.

That lets me compare a sub n and b sub n. Well, how so?

Let me make this assumption that the a sub n's and the b sub n's are non-negative.

I'm going to want that because I'm going to apply the comparison test in a moment.

So I can simplify this a bit.

Instead of making this claim, I can just get

rid of the absolute value bars, it's still true, right.

A sub n over b sub n, minus L is less than L.

You can add L to both sides of that inequality and I get

this, that a sub n over b sub n, is less than 2L.

And now I can multiply both sides by b sub n, and that's okay,

because b sub n is non-negative, so

it doesn't change the direction of this inequality.

And that

tells me that at least for large values of little n, a sub n is less than 2L b sub n.

Now all these pieces are really setting up a comparison test.

How is that going to work?

I got to remember this is only true for large N, but that's okay.

Well suppose that I knew that this series, the sum little

n goes from 1 to infinity of b sub n converged,

well then I would know that this series, the sum little n

goes from 1 to infinity of 2L b sub n also converges.

Right, I can multiply a convergent series just by some number.

But now I'm in the position to apply the comparison test.

Granted this statement's only true for large values of little n but that's

okay, right, because convergence only depends on a tail, so this statement

then implies that this series, the sum of the a sub n's converges, because

this is bigger than a sub n, I mean at least for large values of little n.

So I'm getting a theorem that's telling me if I've got two

series of non-negative terms and this series

converges, then this series converges, provided that

this is true. Let me summarize that.

So, if I've got a sub n's are all non-negative, b sub n's are all

non-negative, the sum of the b sub n's converges, and this limit

statement that the limit of the ratios between the a sub n's and

the b sub n is some finite value L, which is positive, then I

can conclude that this series, the sum of the a sub n's, n goes from 1 to

infinity, converges as well. Now let me exchange the roles of a and b.

So I'd like to be able to start with

the assumption that the series of the a sub n's

converges and then conclude that the series of the b

sub n's converges, but actually that's the exact same statement.

All right, watch this. If I just replace this limit with this

limit, now I'm in the exact same situation, except now b

sub n's and a sub n's roles are exchanged, all right, and that means that if I know

that this series converges, then I know that this series converges as well.

Now put it all together.

Well, since the convergence of a sub n implies the

convergence of b sub n, and the

convergence of b sub n implies the convergence

of a sub n in the presence

of this limit statement, equivalently, this limit statement.

I can simply this a bit, I can say that

if I've got two sequences of numbers, a sub n and

b sub n, both non negative, and this limit exists

and is equal to some number bigger than zero, then this

series converges, if and only if, this series converges.

The sum of the b sub n's converges if and only if, the sum a sub of n's converges.

This convergence test has a name.

This is called The Limit Comparison Test, and it's

one of the situations when two series, in this case

the sum of the b sub n's and the sum of the a sub n's, share the same fate.

They either both converge or they both diverge.

[SOUND]

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