Consider a non-negative integer valued random variable $X$.
Suppose that we are given the generating function $\mathbb{E}[z^X]$, but we do not have it in the form $\sum_{k=0}^\infty\mathbb{P}[X=k]z^k$. That is, we do not know the probabilities $\mathbb{P}[X=k]$.
Is there a way to obtain from the generating function the asymptotic tail probabilities of $X$? That is to say, can we find a sequence $(a_k)$ with $a_k\sim\mathbb{P}[X=k]$ as $k\rightarrow\infty$?