a series related to harmonic series

The series

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{1}{n\sqrt[n]{n}}}={\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{1}{n^{1+\frac{1}{n}}}}$(1)

is divergent.  In fact, since for every positive integer n, one has  $2^n>n$,  i.e.  , any of the series satisfies

$$\frac{1}{n\sqrt[n]{n}}>\frac{1}{2n}.$$

Because the harmonic series and therefore also ${\sum }_1^{\mathrm{\infty }}\frac{1}{2n}$ diverges, the comparison test implies that the series (1) diverges.