Answer
The series diverges by the nth term test.
Work Step by Step
$\Sigma_{n=1}^{\infty}\frac{2n}{3n-2}$
Let's use the nth term test:
If $\lim\limits_{n\to \infty}a_n\ne0$, the series diverges:
$\lim\limits_{n\to \infty}\frac{2n}{3n-2}=\frac{\infty}{\infty}$
$\frac{\infty}{\infty}$ is an indeterminate form, so let's use L'Hopital's rule:
$\lim\limits_{n\to \infty}\frac{2}{3}=\frac{2}{3}$
Since $\frac{2}{3}\ne0$, the series diverges by the nth term test.
