Finding the convergence of a parametric series

Fochina
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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.
 
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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.
 

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