Negative Binomial Series

The Series which arises in the Binomial Theorem for Negative integral $n$,

$$(x+a)^{-n} = \sum_{k=0}^\infty {-n\choose k} x^k a^{-n-k}$$

$$= \sum_{k=0}^\infty (-1)^k {n+k-1\choose k} x^k a^{-n-k}.$$

For $a=1$, the negative binomial series simplifies to

$$(x+1)^{-n}=1-nx+{\textstyle{1\over 2}}n(n+1)x^2-{\textstyle{1\over 6}} n(n+1)(n+2)+\ldots.$$

See also Binomial Series, Binomial Theorem