#### Answer

Diverges

#### Work Step by Step

Given $$\sum_{m=2}^{\infty}\frac{1}{\ln m} $$
Since for $m\geq2$
\begin{align*}
\frac{1}{\ln m} \geq \frac{1}{m}
\end{align*}
Compare with $\displaystyle\sum_{m=2}^{\infty}\frac{1}{m} $, a divergent $p- $series $ (p=1)$; then $\displaystyle\sum_{m=2}^{\infty}\frac{1}{\ln m} $ also diverges.