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

Does it converge?

[NOISE]

There's a somewhat standard process that you can use

to go about checking the convergence of a series.

You got a series.

The sum, n, goes from 1 to infinity a sub n.

And you want to know, does it converge?

I'd recommend applying the limit test first.

Because if you calculate the limit of a sub n as n goes to

infinity, and that's not 0, then you know the series diverges and you're done.

Then,

I try to check absolute convergence.

Right, So if this limits 0 then you don't

know, but you could try to investigate absolute convergence And

you could do that using any of the tests

that we have for series whose terms are not negative.

The root test. The ratio test.

The limit comparison test. What have you.

And if it converges absolutely, you're done.

But if it doesn't converge absolutely,

well, in that case you're back to just looking at the sum of the a sub n's again.

If your series has some positive and some negative terms and it

doesn't converge absolutely, well you better

hope that it's an alternating series.

Because if this is an alternating series, then at least you have the alternating

series test at your disposal. If it's not an alternating

series, well, you could try writing down

the sequence of partial sums, and try to

evaluate the limit just with your bare hands.

[SOUND]

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