Answer
Diverges
Work Step by Step
Let $u_n=\dfrac{2n}{3n-1}$
Now, $\lim\limits_{n \to \infty} u_n =\lim\limits_{n \to \infty}\dfrac{2n}{3n-1}=\lim\limits_{n \to \infty}\dfrac{2}{3-1/n}=(\dfrac{2}{3})$
This means that $\dfrac{2}{3} \ne 0$
Thus, the series diverges due to the n-th term test.