How to test if $\sum_{n=1}^{\infty }\frac{1}{n^{1+\frac{1}{n}}}$ converges? How to test whether $\sum_{n=1}^{\infty }\frac{1}{n^{1+\frac{1}{n}}}$ diverges or converges? Can I apply ratio test? Thanks in advance.

Note that we have $n^{1/n}=e^{\frac1n \log(n)}\le \frac1{1-\frac1n\log(n)}$ for $n\ge 1$.   Therefore, for 
$$\frac{1}{n^{1+1/n}}\ge \frac{1-\frac1n\log(n)}{n}\ge \frac{1-\log(3)/3}{n}$$
Comparison to the harmonic series shows that the series diverges.
