Answer
The series converges for all $r$.
Work Step by Step
Given
$$\sum_{n=1}^{\infty} \frac{r^n}{n!}$$
By using the Ratio Test, we get:
\begin{align*}
\rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty}\left| \frac{r^{n+1}}{(n+1)!}\frac{n!}{r^n}\right|\\
&= \lim _{n \rightarrow \infty} |\frac{r}{n+1}|\\
&= 0<1
\end{align*}
Thus the series converges for all $r$.
