Title: 16_3 The Binomial Series
Course: HELM Workbooks 2023-2024 > Section: HELM 16: Sequences and Series

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

\[1+px+\frac{p(p−1)}{2!}x^2+\frac{p(p−1)(p−2)}{3!}x^3+\cdots \]

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 \(\dfrac{a}{b}\) and multiplying by \(b^n\), the usual binomial theorem:

\[(b+a)^n=b^n+nb^{n−1}a+\frac{n(n−1)}{2!}b^{n−2}a^2+\cdots+a^n.\]

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