In this section, we investigate a specific new type of series that has a variable component.

Section 11.5 Power Series

Definition

A power series is a series of the form:

• x is a variable

• x0 is a specific value of x called the base point

• each ak is a coefficient whose value depends on k

Questions of Interest

For what values of x does the series converge?

To what function does the series converge when it converges?

For Example

This is a geometric series where , and so we know where this converges and to what it converges.

It converges when .

It converges to .

Theorem

Let be a power series.

The set of x for which S converges is an interval centered at x0.

The endpoints of the interval may or may not be included.

The distance from x0 to either endpoint is called the Radius of Convergence, and is denoted R.

R can be 0, any finite positive number, or infinite.

Question

How do we find R?

Question

How do we find R?

If it is a geometric series, we mimic the example from earlier.

If not, we use the ratio test on

Example 1

Find the interval of convergence for the given series and state to what function it converges.

Example 2

Find the interval of convergence for the given series and state to what function it converges.

Example 3

Find the interval of convergence for the given series.

Example 4

Find the interval of convergence for the given series.

Example 5a

Find the interval of convergence for the given series.

Example 5b

Show that:

Example 6

Consider the power series .

(a) Find the domain of f.

(b) Use a partial sum to estimate f(3) within 0.005 of its

actual value.

(c) Use a partial sum to estimate f(-3) within 0.005 of its

actual value.