Following
http://en.wikipedia.org/wiki/Binomial_coefficient#Newton.27s_binomial_series ,
I am trying to prove that
$$ \sum_{\kappa=0}^\infty \binom{\eta + \kappa}{\kappa}x^\kappa = (1 - x)^{-(\kappa+1)}. $$
According to the article, this identity can be derived from the binomial series formula, by applying the identity $$ \binom{n}{k} = (-1)^k\binom{k - n - 1}{k}. $$
To that end, I define $n = \eta + \kappa$ and $k = \kappa$, so that applying the identity, I get
$$ \binom{n}{k} = \binom{\eta + \kappa}{\kappa} = (-1)^\kappa \binom{\eta - \eta - \kappa - 1}{\kappa} = (-1)^\kappa \binom{-\kappa - 1}{\kappa} $$
Now, I do sort of "see", that formally, the binomial series formula should apply. But I don't understand why I am justified in using the formal method. In particular, the binomial coefficients are $0$ when the $n$ term is negative. So I don't really understand the identity
$$ \binom{n}{k} = (-1)^k\binom{k - n - 1}{k}. $$