I'm having trouble evaluating the following sum, for constant $n$ and $-1 \leq x \leq 1$:
$$ \sum\limits_{k = 0}^{\infty} {n+k \choose k}x^k $$
I know the series must converge by the ratio test, since:
$$ \lim\limits_{k \to \infty}\left|\frac{{n+k+1 \choose k+1}x^{k+1}}{{n+k \choose k}x^k}\right| = \lim\limits_{k \to \infty}\frac{n+k+1}{k+1}\left|x\right| = \left|x\right| < 1. $$
However, I'm not sure where to go from here.