# Binomial series

In mathematics, the binomial series is the Taylor series for the function ${\displaystyle f}$ given by ${\displaystyle f(x)=(1+x)^{\alpha }}$ centered at ${\displaystyle x=0}$ where ${\displaystyle \alpha \in \mathbb {C} }$ and ${\displaystyle |x|<1}$. Explicitly,

{\displaystyle {\begin{aligned}(1+x)^{\alpha }&=\sum _{k=0}^{\infty }\;{\binom {\alpha }{k}}\;x^{k}\\&=1+\alpha x+{\frac {\alpha (\alpha -1)}{2!}}x^{2}+{\frac {\alpha (\alpha -1)(\alpha -2)}{3!}}x^{3}+\cdots ,\end{aligned}}}

(1)

and the binomial series is the power series on the right-hand side of (1), expressed in terms of the (generalized) binomial coefficients

${\displaystyle {\binom {\alpha }{k}}:={\frac {\alpha (\alpha -1)(\alpha -2)\cdots (\alpha -k+1)}{k!}}.}$