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

Let's rearrange.

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The alternating harmonic series is a

great example of a conditionally convergent series.

Well, here's the alternating harmonic series.

It's the sum n goes from 1 to infinity of negative 1 to the n plus 1 over n.

And what we know about this is that it

converges by the alternating series test, but it doesn't

converge absolutely, cause if you look at the sum

of the absolute values, you're looking at the harmonic

series, which diverges.

And because it converges but not

absolutely, we call it conditionally convergent.

And what about the negative terms? What if I just add up the negative terms?

Well, here's the first dozen or so terms of the alternating harmonic series.

So 1 over 1 minus a half, plus 1 3rd minus 1 4th plus 1 5th minus 1 6th and so on.

What if I just add up these negative terms?

Just the terms with even index? What do I get?

Well, in that case, what I'm looking at.

Is the sum n goes from 1 to infinity of 1 over 2 n.

And these are the even index terms from the alternating harmonic series.

What do I get if I add up all of these numbers?

Well that's one half of what I get when I add up 1 over n goes from n to infinity.

But, that's half of the harmonic series. Right?

That means that this diverges.

What if I just look at the positive terms?

Well in that case, try to add up 1 over 1,

plus a 3rd, plus a 5th, plus a 7th, plus a 9th.

Alright, I'm trying to figure out just how big is this?

What's the sum n goes from 1 to infinity of the 1 over odd numbers?

Right, 1 over 2n minus 1. Does this

converge or diverge?

Well the trouble is that 1 over 2n minus 1 is even bigger than 1 over 2n.

Right, 1 over 1 is bigger than a half, a 3rd is bigger than a

4th, a 5th is bigger than a 6th, the 7th is bigger than a 8th.

So if this series diverges, then this series diverges as well.

The sum of just the positive terms in the alternating harmonic series diverges.

And yet smoehow that series Has a finite value, so in the alternating

harmonic series, the negative terms diverge, the positive terms diverge.

They both diverge.

So what I've got is really two piles of numbers, and if I take enough

from either pile, I can make a number that's as large as I would like.

This presents me with the following

quite strange opportunity. Here's my goal.

I'm going to rearrange the terms of the

alternating harmonic series to get a new series.

Same terms, just different order, but now my new

series, when I evaluate it, will have value 17.

I'll keep picking up positive terms until I exceed 17.

Okay, well here we go.

Here's the number line, here's zero, here's my goal, 17.

Trying to pick

up numbers from this pile so that I can move all the way past 17.

So I could just get started, right, I pick up one, and that gets me

a little bit closer to 17, I'll pick up the next number, here's a 3rd.

That gets me a bit closer to 17. I'll pick up a 5th.

That gets me a little bit closer to 17.

I'll pick up a 7th, right, and that gets me even a little bit closer to 17.

I mean, the trouble, of course, is that these numbers

are getting smaller, but I know that this series diverges.

So if I keep taking numbers from this pile, I

can move as far to the right as I like.

And indeed, it happens that the sum n goes from 1 to 10 to

the 14th of 1 over 2n minus 1 is a bit bigger than 17.

It's 17.1, so this is how I'm going to start.

I'm trying to write down the same terms as the alternating harmonic series,

but I want a series now that converges to 17.

And this is how I'll start, I'll add 1 over 1 plus a 3rd plus

a 5th all the way to 1 over 2 times 10 to the 14th minus 1.

And that'll land me just to the right of 17.

And now I'll use some of the negative terms.

The sum of these terms from this positive pile was just about 17.1 So

if I pick up a half from the negative pile, and I'm

going to subtract a half now, that moves me over to about 16.6.

Now I'll take some of the positive terms again.

Of course I've already used up a lot of positive terms, but

there's definitely more there that I can add, because this series diverges.

I've only taken away a finite piece of it, so there's more yet to grab.

It's just

the numbers are, are really big. Or really small, rather.

But in any case, there's more terms from this positive pile to add,

and if I add enough of them, I'll eventually move past 17 again.

Maybe I'll end up at, say, 17.001, or thereabouts.

And some more of the negative ones.

I'll pick away this quarter, I'll subtract a

quarter from here, and now my 17.001 or thereabouts

may be moves over to a little bit less than 17, say 16.751.

And I'll just keep on doing this. I can add more positive terms again.

And move myself back to the other side of of 17, and

I'm never going to run out, right, cause this pile of numbers is infinite.

I mean, this series diverges.

So I'm going to keep moving back and forth past 17.

And in the limit, I'll get 17.

So what I mean is that by adding up the same terms.

In the alternating harmonic series, just in a different order.

I'm able to write down a series whose value is 17.

But does that mean that the value of the alternating harmonic series is 17?

No.

No.

We're eventually going to see that the

true value of the alternating harmonic series

Is log 2.

What we're seeing here is the first glimpse of a theorem.

It's a rearrangement theorem.

Here's how it goes. Suppose that L is some real number.

You get to pick out.

And you've got a conditionally convergent series.

In this case, I'm calling it the sum n goes from 1 to infinity of a sub n.

So L's a real number that you picked.

And you're given this conditionally convergent series.

Then that sequence

a sub n can be rearranged to form a new sequence, b sub n.

So the b sub n sequence contains all the terms

of the a sub n sequence, just in a different order.

Well that rearragned sequence b sub n, if you form a series out

of it, the sum n goes from 1 to infinity of b sub n.

The value of that series is L, and you picked L.

What this is saying is that if you're

given a conditionally conversion series, you can rearrange

the terms so that that series sums to any number that you'd like.

In light of this theorem, we have to be careful about how we think about series.

Order matters.

A series is a list of numbers to sum in a given order.

It's not just a pile of numbers that you add up.

The numbers are coming at you in a given order.

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