Answer
The series converges when $|r|<1$
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{n}{n+1}|r|\\\
&= |r|
\end{align*}
Thus the series converges when $|r|<1$.
