Answer
Diverges
Work Step by Step
We will apply the limit comparison test with $a_n=\Sigma_{n=1}^{\infty} \dfrac{1}{{n^{(n+1)/n}}}$ and $b_n=\Sigma_{n=1}^{\infty} \dfrac{1}{n}$
We can see that the series $b_n$ shows a divergent p-series with $p= 1$.
Next, we have $\lim\limits_{n \to \infty} \dfrac{a_n}{b_n} =\lim\limits_{n \to \infty} [\dfrac{\dfrac{1}{{n^{(n+1)/n}}}}{1/n}]$
or, $=\lim\limits_{n \to \infty} n^{-1/n}$
or, $=1$; a finite and positive term
Hence, we can see that the given series diverges by the limit comparison test.
