If series has an infinite radius of convergence then it converges on the whole plane. If the radius of convergence is finite ...

However, even if the radius of convergence is infinite, it does not answer to many questions. F.e. from the decomposition of $e^z$ one cannot derive that $e^x$ fast rapidly as $\mathbb{R}\ni x\to +\infty$ and rapidly decays as $\mathbb{R}\ni x\to -\infty$.