Answer
Diverges
Work Step by Step
$$\lim _{n\rightarrow \infty }\dfrac {n}{\ln n}=\dfrac {\dfrac {d}{dn}\left( n\right) }{\dfrac {d}{dn}\left( \ln n\right) }=\dfrac {1}{\dfrac {1}{n}}=n=\infty \neq 0$$
$$\Rightarrow \sum ^{\infty }_{n=2}\dfrac {n}{\ln n}$$ diverges