Answer
Diverges
Work Step by Step
Let us consider $a_n=\dfrac{\ln n}{n}$
Here, $\ln n \geq 1$ for the all value of $n$.
Apply direct comparison test.
$\Sigma_{n=1}^\infty \dfrac{\ln n}{n} \geq \Sigma_{n=1}^\infty (\dfrac{1}{n})=\Sigma_{n=1}^\infty (\dfrac{1}{n})$
Here, the series $\Sigma_{n=1}^\infty (\dfrac{1}{n})$ shows a harmonic series which diverges.
Thus, the series diverges by the direct comparison test.