Convergence of a parametric serie

Question

I am stuck trying to solve this problem.

I have tu study, for every $a$ real number, the convergence of the serie

$$\sum_{n=1}^{\infty} (\sqrt[n] {n}-1 )^a$$

I tried several ways to solve this problem, with every single criterion for series, but i didn't find nothing.

Can anyone help me?

Thanks in advance!

Answer 1

Hint :

$$\left( \sqrt[n]{n}-1\right)^a \sim \left(\frac{\ln(n)}{n}\right)^a$$