Check whether or not $\sum_{n=1}^{\infty}{1\over n\sqrt[n]{n}}$ converges. Check whether or not $\sum_{n=1}^{\infty}{1\over n\sqrt[n]{n}}$ converges.  
I tried few things but it wouldn't work out. I would appreciate your help.

$$n \cdot n^{1/n} = n e^{\log{n}/n}  = n \left [ 1+ \frac{\log{n}}{n} + O\left ( \frac{\log^2{n}}{n^2} \right ) \right ] = n + \log{n}+ O\left ( \frac{\log^2{n}}{n} \right )  $$
Now,for sufficiently large $n$ (i.e., $n \gt 4$), $n+\log{n} \lt n \log{n} $, so the sum is larger than
$$\sum_{n=2}^{\infty} \frac1{n \log{n}} $$
which diverges.
