### 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