Intuition behind exponential sum convergence My textbook states without proof that the summation:
$$\sum_{x=0}^{\infty} \frac{1}{x!} e^{ax}$$
converges for all real $a$. I am trying to understand this. I assume the reasoning is that the factorial term (which is decreasing in the tail as $x \to \infty$) dominates the exponential (which could be increasing in the tail if $a>0$), but I lack intuition for why the former would dominate. 
I'd appreciate an explanation of any sort -- intuitive, formal, or both.

A formal way to show that this converges is the ratio test: $$\frac{e^{a(x+1)}/(x+1)!}{e^{ax}/x!}=\frac{x!e^{ax}e^{a}}{(x+1)x!e^{ax}}=\frac{e^{a}}{x+1}\to 0<1$$ as $x\to+\infty$.
