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

Alternating series.

[MUSIC]

What is an alternating series? For example, this series the sum

n equals 1 to infinity of minus 1 to the n plus 1 over n,

is an example of an alternating series. Chris, what does that even mean?

Let me write down some of the terms,

because I think will be a little bit clearer.

So if I plug in n equals 1, I get negative 1 squared, which is 1 over 1.

So 1 over 1.

When I plug in n equals 2, I get negative 1 cubed over 2.

That's negative 1 over 2, so minus a half.

When I plug in n equals 3, that's negative 1 to the 4th over 3.

That's 1 over 3.

That's plus a 3rd.

When I plug in n equals 4, I get negative 1 to

the 5th, which is negative 1 over 4, so minus 1 4th.

When I plug in n equals 5,

I get negative 1 to the 6th over 5. That's 1 over 5, so plus a 5th.

When I plug in n equals 6, I get negative 1 to the

7th over 6 so, that's just negative 1 over 6, so minus a 6th.

And then it's going to keep on going like that.

But I'm flip-flopping between these two colors.

I'm alternating sines.

Right?

I'm adding, subtracting, adding, subtracting, adding, subtracting as

an alternating series. So that's what I mean by alternating.

Well here's a precise definition. The series is sum a sub

n, and goes from 1 to infinity,

is called an alternating series,

if a sub n is equal to negative

1 to the nth power times b sub n, and

all the b sub n are the same sign.

So maybe the b sub n sequence, is say all positive, and

then a sub n has this term here, times a positive sequence.

Well this term is what makes the signs, the s i g ns, flip flop back and forth.

Well here's a bit of a warning.

Not every series which has both positive

and negative terms, is an alternating series.

For example this series the sum n goes from 1

to infinity of minus 1 to the n times sign n over n.

This is not an alternating series.

I mean yes, it's got the minus 1 to the

n here, but the sine of n also introduces it's own

quite complicated pattern of positive and negative terms.

So this is a series, some of the terms are

positive, some of the terms are negative, but if not alternating.

Alright, this is not an alternating series you can cook

up other examples sum n goes from 1 to infinity.

Say, minus 1 to the n times n plus 1 over 2.

So that's all in the exponent there.

n times n plus 1 over 2,

say all over n.

This is also not an alternating series, and it's still got a minus 1

there, to raise to some power, but the power is a little bit more complicated.

n times n plus 1 over 2.

terms are some of em will be positive, some are going to be negative,

but they're not flip-flopping in sign, in s i g n back and forth.

In contrast, here's an example that is an alternating series.

the sum n goes from 1 to infinity, say of

minus 1 to the n times sine squared n over n.

Sine squared n, this turns out to always be positive.

Right?

The n is positive here, at minus 1 to the n.

This is the only thing that's affecting the s i g n of this

term, and this does indeed then flip-flop

back and forth, between negative, positive, negative,

positive, negative, positive. This is an alternating series.

[SOUND]

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