Power Series

Definition

A power series centred at is any series of the form:

A Taylor Series is a special case of a power series. The term "Taylor series" implies that Taylor's formula was used. Usually, a series is a Taylor series if it was obtained from a function, whereas a series is called a power series if we use it to define a function.

For example, will give us a Taylor polynomial.

We can easily determine the values of for which a series converges.

Using the ratio test, we will find there is some interval for which the series converges absolutely.

At the points , the series may converge or diverge (we do not know).

We can prove this using the ratio test:

(see limit)

Definition

then the series converges absolutely, and we call the "radius of convergence".

Important

The significance of the interval of convergence is that an absolutely convergent power series behaves just like an ordinary function!

Specifically, if we have two series that converge absolutely in some interval :

then (inside the interval of convergence):

You can add and subtract:
You can multiply and divide
You can take compositions
You can also do calculus

E.g differentiating:

Example

Determine the interval of convergence for .

solution

This is actually a geometric series. It converges if and only if , or , so we conclude that the series converges absolutely on .

Aside: .

Example

Determine the interval of convergence for .

solution
Using the ratio test:

So we have , so the series converges absolutely on .

(see L'Hôpital's rule)

What about endpoints?
For ,

which is an alternating series. , and , so it converges conditionally.

For ,

which is a p-series with , so it diverges.

Example

Determine the interval of convergence for

solution
Note that this is the Taylor polynomial for sine at .

So for all , so the series always converges absolutely.