The Ratio Test will show it converges.
The Ratio Test says:
The series #sum_{n=1}^{\infty} a_n# 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.
