Answer
Absolutely Convergent
Work Step by Step
Let $a_n= \frac{n}{5^n}$. Then $\lim\limits_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right| = \lim\limits_{n \to \infty}\left|\frac{\frac{n+1}{5^{n+1}}}{\frac{n}{5^n}}\right| = \lim\limits_{n \to \infty}\left|\frac{n+1}{5^{n+1}}\cdot\frac{5^n}{n}\right| =\lim\limits_{n \to \infty}\left|\frac{n+1}{n}\cdot\frac{5^n}{5^{n+1}}\right| =\lim\limits_{n \to \infty}\left|\frac{n+1}{n}\cdot\frac{1}{5}\right|= \frac{1}{5}\lim\limits_{n \to \infty}\frac{n+1}{n}=\frac{1}{5}\cdot 1 =\frac{1}{5}.$ Since $\lim\limits_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right| =\frac{1}{5} < 1$, the series ${\displaystyle{\sum_{n=1}^\infty\frac{n}{5^n}}}$ is absolutely convergent by the Ratio Test.