Does the infinite series $\sum_{n=2} \frac{(-1)^n}{\sqrt[n]{ln(n)}}$ converge absolutely / converge / diverge?

Question

Does the infinite series $\sum_{n=2} \frac{(-1)^n}{\sqrt[n]{ln(n)}}$ converge absolutely / converge / diverge?

I can show the the positive and negative element series diverge, so I know the series does not converge absolutely, but I don't know how to tell if it converges.

Answer 1

Note that

$$\frac{1}{\sqrt[n]{\ln n}}\ge \frac{1}{\sqrt[n]{n}}\to 1$$

then the series doesn't converge.

Answer 2

We have $$ \frac{1}{\sqrt[n]{\ln n}} = e^{-\frac{1}{n}\ln\ln n} \xrightarrow[n\to\infty]{} e^0 = 1 $$ so the general term does not converge to $0$: the series diverges by the term test.