Does the series $\sum\limits_{n=1}^\infty \frac{1}{n\sqrt[n]{n}}$ converge?

Does the following series converge? $$\sum_{n=1}^\infty \frac{1}{n\sqrt[n]{n}}$$

As $$\frac{1}{n\sqrt[n]{n}}=\frac{1}{n^{1+\frac{1}{n}}},$$ I was thinking that you may consider this as a p-series with $p>1$. But I'm not sure if this is correct, as with p-series, p is a fixed number, right ? On the other hand, $1+\frac{1}{n}>1$ for all $n$. Any hints ?

Hint: $\sqrt[n]{n}\to1$ when $n\to\infty$ hence, by comparison with the series $\sum\limits_n\frac1n$, this series $______$.

Note that $\sqrt[n]{n}\le 2$. This can be proved by induction, for it is equivalent to $n\le 2^n$.

Thus $$\frac{1}{n\sqrt[n]{n}}\ge \frac{1}{2n}.$$ It follows by Comparison with (half of) the harmonic series that our series diverges.