The application of the Maclaurin series to (1+x)n, for any value of n:
where is a binomial coefficient. The series converges and the expansion is valid in the following cases:(i) for any value of n, if −1<x<1,
(ii) for any value of x, if n is a non-negative integer, in which case the series is finite, since =0 for r>n, and the expansion is that given in the binomial theorem.