Introduction

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

1 + p x + p ( p 1 ) 2 ! x 2 + p ( p 1 ) ( p 2 ) 3 ! x 3 +

We show that this series is only convergent if x < 1 and that in this case the series sums to the value ( 1 + x ) 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 b a , the binomial theorem :

( b + a ) n = b n + n b n 1 a + n ( n 1 ) 2 ! b n 2 a 2 + + a n .

Finally, we use the binomial series to obtain various polynomial expressions for ( 1 + x ) p when x is ‘small’.

Prerequisites

Learning Outcomes