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 Taylor series for the function centered at , where and . Explicitly, where the power series on the right-hand side of () is 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. Closely related is the negative binomial series defined by the Taylor series for the function centered at , where and . Explicitly, which is written in terms of the multiset coefficient Whether () converges depends on the values of the complex numbers α and x. More precisely: If x < 1, the series converges absolutely for any complex number α. If x = 1, the series converges absolutely if and only if either Re(α) > 0 or α = 0, where Re(α) denotes the real part of α. If x = 1 and x ≠ −1, the series converges if and only if Re(α) > −1. If x = −1, the series converges if and only if either Re(α) > 0 or α = 0. If x > 1, the series diverges, unless α is a non-negative integer (in which case the series is a finite sum). In particular, if is not a non-negative integer, the situation at the boundary of the disk of convergence, , is summarized as follows: If Re(α) > 0, the series converges absolutely. If −1 < Re(α) ≤ 0, the series converges conditionally if x ≠ −1 and diverges if x = −1. If Re(α) ≤ −1, the series diverges. The following hold for any complex number α: Unless is a nonnegative integer (in which case the binomial coefficients vanish as is larger than ), a useful asymptotic relationship for the binomial coefficients is, in Landau notation: This is essentially equivalent to Euler's definition of the Gamma function: and implies immediately the coarser bounds for some positive constants m and M .