The Ratio Test will show it converges.
The Ratio Test says:
The series sum_{n=1}^{\infty} a_n∞∑n=1an converges if \lim_{n \to \infty} \abs\frac{a_{n+1}}{a_n} < 1, diverges if \lim_{n \to \infty} \abs\frac{a_{n+1}}{a_n} > 1, and is inconclusive if \lim_{n \to \infty} \abs\frac{a_{n+1}}{a_n} = 1.
For your specific problem, a_n = \frac{n}{3^{n+1}} and a_{n+1} = \frac{n+1}{3^{n+2}} so
\lim_{n \to \infty} \abs\frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \abs\frac{\frac{n+1}{3^{n+2}}}{\frac{n}{3^{n+1}}} =
= \lim_{n \to \infty} \abs{\frac{n+1}{3^{n+2}} \cdot \frac{3^{n+1}}{n}} = \lim_{n \to \infty} \abs{\frac{n+1}{3n} } =
= \frac{1}{3}\lim_{n \to \infty} \abs{\frac{n+1}{n} } = \frac{1}{3} < 1, so the series converges.
