# Binomial Series

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

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

This series expansion is:

$\begin{array}{l}{\left(1+x\right)}^{\alpha }={\sum }_{k=0}^{\infty }\left(\begin{array}{c}\alpha \\ k\end{array}\right){x}^{k}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=1+\alpha x+\frac{\alpha \left(\alpha -1\right)}{2!}{x}^{2}+...\end{array}$

If $|\text{\hspace{0.17em}}x\text{\hspace{0.17em}}|<1$ , the series converges. If $|\text{\hspace{0.17em}}x\text{\hspace{0.17em}}|>1$ , the series diverges.