This is a rather obvious question, but my Google-fu is failing me.
Given that $F(z) = \sum_{n=1}^{\infty} a_n z^n$, can one say anything about $G(z) = \sum_{n=1}^{\infty} \frac{1}{a_n} z^n$?
In particular, I'm looking for a solution to this question using generating functions to get asymptotics for the coefficients, but I've only ever seen generating functions applied where there's an obvious way to manipulate the coefficients to match up with the recursion. It might not be possible to do this with generating functions, but I'd love to see it done.