Binomial series

In mathematics, the binomial series is a generalization of the binomial formula to cases where the exponent is not a positive integer:

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

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where ${\displaystyle \alpha }$ is any complex number, and the power series on the right-hand side is expressed in terms of the (generalized) binomial coefficients

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

The binomial series is the MacLaurin series for the function ${\displaystyle f(x)=(1+x)^{\alpha }}$. It converges when ${\displaystyle |x|<1}$.

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). In this case, the series is a finite polynomial, equivalent to the binomial formula.