I would like to calculate the summation of an infinite series, where each term looks like a binomial object. However each binomial term has a different "probability" in the series.
$$\sum_{x=1}^\infty{N \choose k}(f(x))^k(1-f(x))^{N-k}\\f(x)+(1-f(x))=1$$ where $f(x)$ is some arbitrary function on x.
If for instance, $f(x)=x^{-a}$. Is there a straightforward way to do this series summation? In the limit, the terms go to 0 so this series should converge. But I can't see a way to do the series summation.
I was thinking of trying a generating function approach. But I don't have much experience with manipulating them, so I am stuck there.