# what is the region of convergence of this exponential?

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.

• It converges for all $x\in \mathbb C.$ See: Hadamard Radius Formula. – DanielWainfleet Apr 8 '17 at 13:12

The easiest way to see that it converges for all $x$ is that for all $u>2|x/a|$ we have $$|(x/a)|^{u+1}/(u+1)!\leq \frac {1}{2}|x/a|^u/u!.$$