Binomial Series

The binomial series is the Talyor series (or Maclaurin series) of the function

${\left(1+x\right)}^{\alpha }$

This series expansion is:

$\begin{array}{l}{\left(1+x\right)}^{\alpha }={\displaystyle {\sum }_{k=0}^{\infty }\left(\begin{array}{c}\alpha \\ k\end{array}\right)x^k}\\ =1+\alpha x+\frac{\alpha \left(\alpha -1\right)}{2!}{x}^2+\mathrm{...}\end{array}$

If $\left|x\right|<1$ , the series converges. If $\left|x\right|>1$ , the series diverges.