Answer
Converges
Work Step by Step
Given $$\sum_{n=1}^{\infty}\frac{n}{3^{n}} $$ Since for $n\geq1$ \begin{align*} n &\leq 2^{n}\\ \frac{n}{3^{n}} &\leq \frac{2^{n}}{3^{n}}\\ \frac{n}{3^{n}}& \leq\left(\frac{2}{3}\right)^{n} \end{align*} Compare with $\displaystyle\sum_{n=1}^{\infty} \left(\frac{2}{3}\right)^{n}$, a convergent geometric series $|r<1| $; then $\displaystyle\sum_{n=1}^{\infty}\frac{n}{3^{n}} $ also converges
