Is this series $\sum_{n\geq 2} (\ln n)^{-\sqrt{n}} $ convergent?

Question

Determine whether the following series is convergent or divergent: $$\sum_{n\geq 2} (\ln n)^{-\sqrt{n}}.$$

Please give me some hints!

Answer 1

Hint. Compare the term with $1/n^2$ by evaluating the limit $$\lim_{n+\infty}\frac{n^2}{(\ln n)^{\sqrt{n}}}= \lim_{n+\infty}\exp(2\ln(n)-\sqrt{n}\ln(\ln n))).$$

Answer 2

An idea: For large $n,$ $(\ln n)^{-\sqrt n} < 2^{-\sqrt n}.$ Now

$$\tag 1\sum_{n=1}^{\infty}2^{-\sqrt n} = \sum_{m=1}^{\infty}\sum_{n=m^2}^{(m+1)^2-1}2^{-\sqrt n}.$$

In the inner sum on the right there are $2m+1$ terms, each of which is bounded above by $2^{-m}.$ Thus the right side of $(1)$ is no more than

$$\sum_{m=1}^{\infty}(2m+1)2^{-m}.$$

Now we're down to a sum that we can understand with standard tools.