TITLE: Is the series $\sum_{k=1}^{\infty}(\sqrt[k]{k}-1)$ divergent or convergent?
QUESTION [5 upvotes]: I find this series to be a bit troublesum. I can't find a proper method that works. Root test doen't seem to be useful. Ratiotest got ugly. Integral test seems to force me to integrate a function $x^{1/x},$ which doesn't have any elementary primitive.
I can't find an apropriate limit to compare it to and do a limit test comparison.

REPLY [7 votes]: We have $$k^{1/k} -1= e^{\frac{\ln(k)}{k}}-1 >\frac{\ln(k)}{k},$$ so by comparison, the sum diverges.