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

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

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.

Whether (1) converges depends on the values of the complex numbers α and x. More precisely:

In particular, if α is not a non-negative integer, the situation at the boundary of the disk of convergence, |x| = 1, is summarized as follows:

The following hold for any complex number α:

Unless ${\displaystyle \alpha }$ is a nonnegative integer (in which case the binomial coefficients vanish as ${\displaystyle k}$ is larger than ${\displaystyle \alpha }$), a useful asymptotic relationship for the binomial coefficients is, in Landau notation: