### Introduction

In this Section we examine an important example of an infinite series, the binomial series:

$\phantom{\rule{2em}{0ex}}1+px+\frac{p\left(p-1\right)}{2!}{x}^{2}+\frac{p\left(p-1\right)\left(p-2\right)}{3!}{x}^{3}+⋯$

We show that this series is only convergent if $\left|x\right|<1$ and that in this case the series sums to the value ${\left(1+x\right)}^{p}$ . As a special case of the binomial series we consider the situation when $p$ is a positive integer $n$ . In this case the infinite series reduces to a finite series and we obtain, by replacing $x$ with $\frac{b}{a}$ , the binomial theorem :

$\phantom{\rule{2em}{0ex}}{\left(b+a\right)}^{n}={b}^{n}+n{b}^{n-1}a+\frac{n\left(n-1\right)}{2!}{b}^{n-2}{a}^{2}+⋯+{a}^{n}.$

Finally, we use the binomial series to obtain various polynomial expressions for ${\left(1+x\right)}^{p}$ when $x$ is ‘small’.

#### Prerequisites

• understand the factorial notation
• have knowledge of the ratio test for convergence of infinite series.
• understand the use of inequalities

#### Learning Outcomes

• recognise and use the binomial series
• state and use the binomial theorem
• use the binomial series to obtain numerical approximations