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

Infinite radius.

It can certainly happen that the radius of convergence is infinite.

Well, let's consider this power series.

The sum, n goes from 0 to infinity, of x to the n

divided by n factorial. Let's try the ratio test.

So, here we go.

the limit n goes to infinity of the n plus first term.

So, x to the n plus 1

over n plus 1 factorial divided by just

the nth term which is what's displayed here.

X to the n over n factorial and absolute value of

it 'because I'm checking for absolute convergence of the ratio test.

I can simplify this.

If it is a fraction with fractions in the

numerator and denominator, this is the limit n goes

to infinity of x to the n plus 1

times n factorial in the denominator of the denominator,

put that in the numerator.

Divided by x to the n times n plus 1 factorial.

So, i've just got a fraction, but I can simplify this

a bit too, this is the limit n goes to infinity.

We've got x to the n plus 1 over x to the n.

Which just leaves me with an x in the numerator, and I've got

n factor on the numerator, and an n plus 1 factorial in the denominator.

Well, n plus 1

factorial kills everything here, right?

N plus 1 factorial is 1 times 2, all the way through

n plus 1, and that contains all the terms in n factorial.

So, what I'm left with, is just an n plus 1 and the denominator.

Now, what is this limit?

X is just some fixed quantity, it doesn't depend on n.

But what's the limit then, of some number x

divided by n plus 1, n going to infinity?

Well, this limit is zero. Alright?

It doesn't matter what x is.

If you take some fixed number and divide it by a very

large quantity, you can make this as close to zero as you'd like.

So, this limit is zero. Zero is less than one.

And that means by the ratio test, this series converges regardless of what x is.

So, what's the radius of convergence? Series converges for

all values of x.

And that means the radius of convergence is infinity.

And in the not to distant future, we're going to see a very surprising result.

We're eventually going to see that this series.

Is in fact a complicated way of writing down a function we already know.

This is just a complicated way of writing down the function e to the x.

Meaning that if you plug in specific values

for x, say x equals negative 1, you'd get

that the sum n goes from 0 to infinity of minus 1 to the n over n factorial.

Is equal to e to the negative 1, it's equal to 1 over e.

And by choosing different values of x, by, by plugging in different

specific values of x, we can generate a ton of really neat series.

Like, here's another example.

I mean, just the fact that the sum, n goes from 0 to

infinity of 10 to the n over n factorial, is, well, according to this.

That's just e to the 10th power. And, that is really cool.

But, let me warn you.

And, share a bit of the philosophy of the power series with you.

Yes, by plugging in specific values of x,

you can generate a ton of interesting examples.

But, power series aren't just a way

of generating a bunch of series and isolations.

Part of the joy of power series comes by thinking of power

series, not as a, a mechanism for generating a bunch of discrete examples.

But as a way of collecting together a whole

bunch of interesting series that depend on a parameter x.

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