So I am doing some taylor series which boil down to the geometric series, for which I then need to evaluate various binomial coefficients. I have always used $n\choose k$ $=\frac{n!}{(n-k)!k!}$ to do these, this however becomes troublesome for non-integer and/or negative values of n.
Now I just memorized that if $k=0$, $n\choose k$ $= 1$ for integers $ n\geq 0$. But this still leaves me with negative and/or negative values for n. My textbook provides me with $n\choose k$ = $\frac{n(n-1)...(n-k+1)}{k!}$, but I`m still not sure how to evaluate this with $k=0$. Somehow I always get it wrong.
For example i get $-1\choose 0$ $=\frac{-1(-1-0+1)}{0!}=\frac{-1*0}{1}=0$, but it should be $1$. Whats my error? Thanks!