Answer
Divergent
Work Step by Step
By Root Test, we have
$\lim_{n\to\infty} \sqrt[n]{ \biggl|\frac{n}{\ln n}\biggr|^n}=\lim_{n\to\infty}\frac{n}{\ln n} $
We can throw the absolute sign since $n \geq 2\implies\frac{n}{\ln n} > 0$
And since the limit is indeterminate form $\frac{\infty}{\infty}$, we can apply L'Hopital's Rule,
$\lim_{x\to\infty}\frac{x}{\ln x}=\lim_{x\to\infty}x=\infty$
The above equation implies
$\lim_{n\to\infty}\frac{n}{\ln n } = \infty$
Since, the limit is $> 1$, the series is divergent
