Answer
Divergent
Work Step by Step
If a series is convergent, then $\lim\limits_{n \to \infty}a_{n}=0$
$\lim\limits_{n \to \infty}a_{n}=\lim\limits_{n \to \infty}ln(\frac{n}{3n+1})$
$=ln\frac{1}{3}\ne 0$
Hence, the series is divergent since the limit does not equal to $0$, by the Test for Divergence.
