Is $\sum\frac1{p^{1+ 1/p}}$ divergent?

Is $\displaystyle\sum\frac1{p^{1+ 1/p}}$ divergent? How can we prove that it is divergent or convergent in analytic number theory? I know what bound of the n-th prime number is, and that its order is $n\log(n)$. Maybe we can use the divergence of $\displaystyle\sum\frac1{n^{1+ 1/n}}$ to show that. I'm not sure that $\displaystyle\sum\frac1{n^{1+ 1/n}}$ is divergent, but I think it is. So would you please help me with this ? Can you help me in finding a proof for it ? Thank you very much, friends.

Solution 1:

This is a great question. But you can show that $$\frac 1{p^{1+1/p}} > \frac 1{2p}$$ since $2^p >p$, and thus your series diverges.