Binomial series

In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like for a nonnegative integer . Specifically, the binomial series is the MacLaurin series for the function , where and . Explicitly,

(1)

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

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