Question: Given some sequence of coefficients (either explicitly or by way of a recursion formula) for a power series, is it possible to determine the end behavior (i.e. convergence, bounded, unbounded, etc.) of the that power series?
The Maclaurin series for both $e^x$ and $e^{-x}$ have an infinite radius of convergence, but the latter converges to $0$ as $x\rightarrow\infty$ while the former diverges in the same limit. Likewise, the Maclaurin series for $\cos(x)$ and for $e^{-x^2}$ are very similar (e.g. both are even and alternating, and both divide by factorials), but while they both stay bounded, $\cos(x)$ does not converge as $x\rightarrow\pm\infty$ whereas $e^{-x^2}$ does.
It would appear that alternating is a necessary condition for convergence to $0$ (I certainly haven't found any counter-examples), but it is clearly not a sufficient condition.
I'm particularly interested in this question as it applies to series solution of differential equations.
It is not quite possible to see the asymptotic at $+\infty$ of an entire function just by looking at the asymptotic of its coefficients. If $f(x)\to 0$ then $f(x)+\epsilon x^n$ grows like a polynomial whereas the coefficients of both are quite similar.