Suppose $f(z)=\displaystyle\sum_{n=0}^\infty a_nz^n$ has infinite radius of convergence (where the $a_n\in\mathbb{C}$). Is there any tests which can be done on the sequence $\{a_n\}$ to determine whether $f$ converges to zero along the real line? That is, whether $\displaystyle\lim_{x\to\pm\infty}f(x)=0$?
Necessary or sufficient conditions are welcome.