# Binomial Theorem for Negative Index and Negative Parameter

## Theorem

Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $z \in \R$ be a real number such that $\left\lvert{z}\right\rvert < 1$.

Then:

 $\displaystyle \dfrac 1 {\left({1 - z}\right)^{n + 1} }$ $=$ $\displaystyle \sum_{k \mathop \ge 0} \binom {-n - 1} k \left({-z}\right)^k$ $\displaystyle$ $=$ $\displaystyle \sum_{k \mathop \ge 0} \binom {n + k} n z^k$

where $\dbinom {n + k} n$ denotes a binomial coefficient.