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

Radius zero.

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Nobody's guaranteeing us all that much convergence.

As an example, let's consider this power series.

The sum n goes from 0 to infinity.

Of n factorial times x to the n.

For sure, if x is equal to 0, then this power series converges.

Yeah, so what happens when x is equal to 0?

Then this series is the sum n goes from 0

to infinity, of n factorial times 0 to the nth power.

Now what is this sum?

Right, does this series converge?

Well, I prefer the convention that 0 to the 0 is 1.

And I think pretty much everyone prefers the convention that

0 factorial is 1.

So the first term, the n equals 0 term of this series is equal to 1.

But what about the next term? What about the n equals 1 term?

Well, that's 1 factorial times 0 to the first power, that zero.

What about the next term?

What about n equals 2?

That's 2 factorial times 0 squared. That's 0.

What about n equals 3?

That's a number times 0 to the third power, that's zero.

What about n equals 4?

That's a number times 0 to the fourth power.

That's

[INAUDIBLE].

All the other terms in this series are 0. So this series has the value 1.

It, it converges at x equals 0. But are there any

other values of x, besides 0, for which this series converges.

So I know this converge at x equals 0.

But lets suppose that x is some non-zero number.

Then does this series converge or diverge.

Well, let's check it with the ratio test.

So, I'm going to look at the limit as n goes to infinity.

Once the n plus first term here, that's n plus 1 factorial times x to

the n plus 1 divided by just the nth term, which is n factorial times x to the n.

Look at the absolute value of that.

What's this limit? Well I can simplify this limit, right?

This is the limit of what's n plus 1 factorial over n factorial?

Well I can simplify a bit. That's just n plus 1.

And then I've got x to the n plus 1 over x to the n, that's just x.

So for some fixed value of x which isn't 0.

What is this limit?

Well if x is anything but 0, this limit is enormous number, times non-zero number.

This is very, very positive, right? This limit is infinite.

And that is bigger than 1.

What that means is that by the ratio test, this series diverges.

For any fixed value of x not 0, this series diverges.

So it doesn't converge anywhere else, it only converges when x equals 0.

So in other words, since this series only converges at the

point x equals 0, and diverges whenever x is not zero.

That means the radius of convergence, is equal to 0.

Let me leave you with a question.

Try to think of other power series with radius of convergence equal to 0.

Can you think of any other examples?

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