Finding the convergence of a parametric series

[Fochina]

Homework Statement

find for what $\alpha$ the series converges

Relevant Equations

$$\sum_{n}\left ( \sqrt[n]{n}-\sqrt[n]{2} \right )^\alpha $$

It is clear that the terms of the sequence tend to zero when n tends to infinity (for some α) but I cannot find a method that allows me to understand for which of them the sum converges. Neither the root criterion nor that of the relationship seem to work. I tried to replace $\sqrt[n]{n}$ with $e^{\frac{ln n}{n}}$ but then I don't understand how to proceed.

 

[fresh_42]

The result seems to be $\alpha \geq 1$. So you could work with a convergent upper and a divergent lower bound which gives you some flexibility to change the function.