sequences and series - How to evaluate this limit and its convergence? $sum_{n=1}^inftyfrac{1}{nsqrt[n]{n}}$

How to evaluate this limit
$$\sum_{n=1}^\infty\frac{1}{n\sqrt[n]{n}}$$

and its convergence?

I tried ratio test, root test, Raabe's test. However, I'm not getting anywhere. Can you please help me? Thank you

Answer

For $n$ sufficiently large, $\root n\of n<2$; so,
$${1\over n\,\root n\of n}>{1\over 2n}$$ for sufficiently large $n$.

Since the series $\sum\limits_{n=1}^\infty {1\over 2n}$ diverges (it is essentially the harmonic series), it follows from the Comparison test that the series $\sum\limits_{n=1}^\infty {1\over n\,\root n\of n}$ diverges.