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 tail is what matters.

[MUSIC]

If you go out and read the literature

on convergence tests and the like, you'll sometimes find

that those theorems are written in a way

that's a bit more vague than we're used to.

The Comparison Test.

If a sub n is between 0 and b sub n for all n, and the series sum of the

b sub n's converges, then this series, the sum of the a sub n's converges as well.

Do you see what's missing? Well, I didn't write the sum n goes from 1

to infinity of b sub n, or the sum n goes from

1 to infinity of a sub n. But this is actually okay.

here's a theorem.

Let M be a natural number then this series, the

sum m goes from M to infinity of a sub m,

converges, if and only if this series, the sum little m goes from 1 to

infinity of a sub m, converges. In short, convergence depends on the tail.

So I've got this series, the sum little m goes from 1 to infinity of a sub m.

And I could start writing it out like this.

It's a sub 1 plus a sub 2, blah, blah, blah,

plus a sub M minus 1 plus a sub M plus a

sub M plus 1 plus and so on. And I can take out my scissors of

mathematics and cut this series like this.

This piece here I'll call the head, and what's left over, which is really the

sum m goes from M to infinity of a sub m, this is the tail.

And what the theorem tells me is that this series converges if and

only if this series converges So converges only depends on the tail.

Why does convergence only depend on the tail?

Well, let's suppose that this tail converges, that

means something of the sequence of partial sums.

What that means is, that if I consider the

sequence of partial sums, s sub n equals the sum

m from M to n of a sub m. That sequence of partial

sums converges to something, I'll call it L, it's the value of this series.

I can relate those partial sums to the partial

sums for the series that begins with a sub 1.

So now let's think about this series.

Right, this series m goes from one to infinity of a sub m.

The value

of this series is by definition the limit of the partial sums for that series, and

that's the limit n goes to infinity of the sum little m goes from 1 to n of a sub m.

Now I can analyze this in terms of the series that starts at M.

Alright?

What is this limit?

Well, that limit is a sub 1 plus dot, dot, dot, plus a sub M minus

1 plus this limit.

The limit n goes to infinity of the sum n from M to n of a sub M.

Alright, both of these are just the limits of the

sum of the terms, a sub 1 through a sub n.

But now, I'm assuming that this limit exists because I'm assuming

that that, that series that starts at M converges as the limit

of the partial sums for that series which I was calling L before, which is just to

say that this series has a finite value, right, this limit exists.

And what that means, is that if I suppose that the tail

of a series converges, then the original series converges as well.

Here's the upshot. If you only care about

convergence, writing down this, just the sum over n

of a sub n, is just as good as writing

down this more complicated looking thing, the sum little

n goes from 1 to infinity of a sub n.

Because if all I care about is convergence,

I don't actually need to know where the series

is starting, because regardless of where I start

the series, they all converge or they all diverge.

It only depends on the tail.

[SOUND]

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