### Introduction

In this Section we consider power series. These are examples of infinite series where each term contains a variable,
$x$
, raised to a positive integer power. We use the ratio test to obtain the
**
radius of convergence
**
$R$
, of the power series and state the important result that the series is absolutely convergent if
$\left|x\right|<R$
, divergent if
$\left|x\right|>R$
and may or may not be convergent if
$x=\pm R$
. Finally, we extend the work to apply to general power series when the variable
$x$
is replaced by
$\left(x-{x}_{0}\right)$
.

#### Prerequisites

- have knowledge of infinite series and of the ratio test
- have knowledge of inequalities and of the factorial notation.

#### Learning Outcomes

- explain what a power series is
- obtain the radius of convergence for a power series
- explain what a general power series is

#### Contents

1 Power series2 The radius of convergence

3 Properties of power series

4 General power series