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

Taylor Series

In this last module, we introduce Taylor series. Instead of starting with a power series and finding a nice description of the function it represents, we will start with a function, and try to find a power series for it. There is no guarantee of success! But incredibly, many of our favorite functions will have power series representations. Sometimes dreams come true. Like many dreams, much will be left unsaid. I hope this brief introduction to Taylor series whets your appetite to learn more calculus.

- Jim Fowler, PhDProfessor

Mathematics

Welcome to week six of Sequences and Series.

[MUSIC]

Well thus far, we've been starting with a power series, and

then asking the question, what function does that power series represent?

For example, we considered this series, the sum n goes from 0 to infinity of

x to the n and we ended up showing that this series is equal to 1 over 1 minus x.

Provided that the absolute value of x is less than 1.

We've also built a lot of

tools for transforming one power series into a new power series.

For instance, I can differentiate a power series term by term.

So in this case, what do I get?

Well, if I differentiate this term by term, I get the sum n

goes from 1 to infinity of n times x to the n minus 1.

That's the derivative of x to the n with respect to

x, and that's the derivative of 1 over 1 minus x, and

we calculated the derivative of 1 over 1 minus

x, well, that's 1 over 1 minus x squared.

So there we've got it, I've got a

power series, and I found a function that represents

that power series at least on this interval when

the absolute value of x is less than 1.

So that's what we've been doing, we've been starting with the power series and

we've been trying to identify, you know,

what function does that power series represent.

And then maybe we'd be transforming

those power series, messing around with them by differentiating them term by

term, integrating them term by term,

multiplying them together, things like that.

But this week, we're going to turn all that around.

What I mean is that so far in the course,

we've been starting with a power series and then from

that power series we've been getting an iso description of

the power series as some function that we're familiar with.

And, I want to turn that around now.

I want to start with the description of a nice function and then

try to find a power series representing that function on some interval.

[SOUND]

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