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

Power Series

In this fifth module, we study power series. Up until now, we had been considering series one at a time; with power series, we are considering a whole family of series which depend on a parameter x. They are like polynomials, so they are easy to work with. And yet, lots of functions we care about, like e^x, can be represented as power series, so power series bring the relaxed atmosphere of polynomials to the trickier realm of functions like e^x.

- Jim Fowler, PhDProfessor

Mathematics

Convergence depends on x.

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Let's consider a power series, so maybe I've got a power series,

the sum, n goes from zero to infinity of some coefficients,

a sub n, times x to the nth power.

Now suppose that I know that that power series converges absolutely,

when x equals three.

So yeah, I'm gonna assume that it converges absolutely at x=3, and

I want to know what about our other points.

What about at, say x=2?

Does it converge there?

Well then it must converge when x=2.

Let's see why.

Well since it converges absolutely at x=3,

that just means that the sum, n goes from zero to infinity,

of the absolute value of a sub n, times 3 to the n, right?

This series converges.

We can compare this to the same series when x equals two.

What I mean to say, is just that zero is less than or

equal to the absolute value a sub n, times 2 to the n, which is less than or

equal to the absolute value of a sub n, times 3 to the n.

And since this series, the sum of a sub n,

times 3 to the n converges, that means by the comparison test,

this series, the summing goes from zero to infinite of a sub n,

times 2 to the n, this series converges.

Which is just to say that the original series, when x=2.

Well, in that case, this series converges absolutely.

Of course, there's nothing special about the number two.

So if x is any value, so

the absolute value of x is less than or equal to 3, that just means that x

is in the interval from -3 to 3.

If x is any value in that interval, then zero is less than or

equal to the absolute value of a sub n, times x to the n.

Well that's just cuz it's the absolute value of something.

But then that is less than or

equal to the absolute value of a sub n, times 3 to the n.

So again, by comparison, right, that means that this series,

the sum n goes from zero to infinity, I'll just write a sub n, x to the n,

converges absolutely, because the sum of the absolute value converges,

because I'm comparing with this convergence series.

And this is the usual case, this is usually what happens.

To talk about this though, let me be a little bit more formal.

Let me give a name to this, let's call C the collection of all real numbers,

so that this power series converges.

Or in words, C is all the real numbers x so that this series converges.

It's a collection of numbers.

The big deal here is that C is an interval.

Well, here's the theorem.

This collection of values of x where the power series converges,

it turns out that collection of points is an interval,

by which I mean maybe it's this open interval, maybe it's a closed interval, or

maybe it's something more complicated, like some half open interval.

We'll see a proof of that soon.

And since it's an interval, this collection of points,

where the power series converges, is called the interval of convergence.

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