Section 10.14

Power Series

Series that are functions of . The foundation of approximation.

Introduction

A Power Series about

is a series of the form:

It defines a function

. But where is this function valid?

Interactive: Radius of Convergence

For , the Radius . Notice how the blue sum diverges wildly outside .

Radius of Convergence

For any power series, there are exactly three possibilities:

Converges only at (Radius ).
Converges for all (Radius ).
Converges for (Radius is a finite number).

Interval of Convergence

The interval is . But what about the endpoints? You MUST check and individually using other tests.

Step-by-Step:
1. Use Ratio Test on to find .
2. Set to find the open interval for .
3. Plug in endpoints into original series and check convergence.

Worked Examples

Example 1: Finite Radius

Find interval for .

1. Ratio Test:
.
Set .
Radius . Interval .

2. Check Endpoints:
(Div harmonic).
(Conv alt harmonic).

Interval: .

Example 2: Infinite Radius