Binomial series

In mathematics, the binomial series is the Maclaurin series for the function ${\displaystyle f}$ given by ${\displaystyle f(x)=(1+x)^{\alpha }}$, where ${\displaystyle \alpha \in \mathbb {C} }$ is an arbitrary complex number. Explicitly,

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

If α is a nonnegative integer n, then the (n + 2)th term and all later terms in the series are 0, since each contains a factor (n − n); thus in this case the series is finite and gives the algebraic binomial formula.

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