Answer
Converges
Work Step by Step
Let us consider $a_n=\dfrac{1}{ n3^{n}}$and $u_n$ refers to all positive values of $n$.
Here, $f(n)=\dfrac{1}{(n)(3^{n})}$; $f'(n)=\dfrac{-3^n(1+\ln 3)}{(n^2)(3^{2n})} \lt 0$
The negative sign simplifies that the sequence $u_n$ is not increasing.
Then, $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty}\dfrac{1}{(n)(3^{n})}=0$
Hence, the series converges by the Alternating Series Test.