Answer
Converges
Work Step by Step
Given
$$\sum_{n=1}^{\infty} \frac{(-1)^{n-1} n}{5^{n}}$$
By using the ratio test
\begin{align*}
\rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty}\left|\frac{\frac{1}{5} \times \frac{(-1)^{n}(n+1)}{5^{n}}}{\frac{(-1)^{n-1} n}{5^{n}}}\right|\\
&=\lim _{n \rightarrow \infty} \frac{n+1}{5n}\\
&=\frac{1}{5}<1
\end{align*}
Thus the series converges.
