Given a sequence $a_n$, what can be said about the limiting behavior of $f(z) = \sum_{n=0}^\infty \frac{a_n}{n!} z^n$ as $z \rightarrow \pm\infty$? This seems like a pretty natural question to ask, so can anyone point me to some resources discussing it?
I've wondered about this for a long time, but for my particular application, I have a sequence $a_n = c + O(b^n)$ for some $b\in(0,1)$, and I'm wondering if anything can be determined about the the behavior of $f(z)$ as $z$ goes to $-\infty$.
EDIT: as I can see from the comment section knowing the asymptotic behavior of $a_n$ is not enough to know what happens to $f$ as $z\rightarrow-\infty$. Is there any way by looking at $a_n$ to determine the asymptotic behavior of $f$ or is is a lost cause?