Binomial series

In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like ${\displaystyle (1+x)^{n}}$ for a nonnegative integer ${\displaystyle n}$. Specifically, the binomial series is the MacLaurin series for the function ${\displaystyle f(x)=(1+x)^{\alpha }}$, 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)

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

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

Note that if α is a nonnegative integer n then the x^{n + 1} term and all later terms in the series are 0, since each contains a factor of (n − n). Thus, in this case, the series is finite and gives the algebraic binomial formula.