what is the region of converge for the series expansion of the exponential: $\exp(-\frac{x}{a})$, where: $x$ is a positive variable and a is positive number.
i'm using its series representation, which is : $\sum_{u=0}^{\infty}\frac{(-1/(a))^u}{u!} x^u$.
i'm using this expansion in an integral to be able to evaluate it, i'm afraid it will make problems due to the region of convergence.
The easiest way to see that it converges for all $x$ is that for all $u>2|x/a|$ we have $$|(x/a)|^{u+1}/(u+1)!\leq \frac {1}{2}|x/a|^u/u!.$$