Answer
The given series is divergent.
Work Step by Step
$\Sigma^{\infty}_{n=1} \frac{n-1}{3n-1}$
$\lim\limits_{n \to \infty}a_{n} = \lim\limits_{n \to \infty}(\frac{n-1}{3n-1})$
$= \lim\limits_{n \to \infty} (\frac{1-\frac{1}{n}}{3-\frac{1}{n}})$
$= \frac{1}{3} \ne 0$
The given series diverges since $\lim\limits_{n \to \infty}a_{n} \ne 0$. The series $\Sigma^{\infty}_{n=1} a_{n}$ is divergent.
