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 $|x| , divergent if $|x|>R$ and may or may not be convergent if $x=±R$ . 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