Answer
Diverges
Work Step by Step
We know that for all $\ln n < n$ for all $n\geq 2$.
We have now $\frac{1}{\ln n}>\frac{1}{n}$ since the left side has a smaller denominator.
Since $\sum_{n=2}^\infty \frac{1}{n}$ is a divergent $p-$series with $p=1$, it follows by the Direct Comparison Test with $a_n=\frac{1}{\ln n}$ and $b_n=\frac{1}{n}$ that the series $\sum_{n=2}^\infty\frac{1}{\ln n}$ diverges.
