What is the analytic representation of $\sum_{n=0}^\infty \frac{ a^nx^n}{n!}$?

Question

I only know that a geometric series is $$\frac1{1-x}\qquad (|x|<1)$$ and this looks similar.

Answer 1

We have

$$ e^{z}=\sum_{n=0}^{\infty}\frac{z^n}{n!},\quad z \in \mathbb{C}. $$

You may put $z:=a x$ then $z^n=a^nx^n$ giving $$ \sum_{n=0}^{\infty}\frac{a^n x^n}{n!}=e^{ax}. $$