Convergence of $\sum_{n=1}^\infty \frac{\sqrt[n]{n} - 1}{n}$ 
I am looking for the convergence of
$$\sum_{n=1}^\infty \frac{\sqrt[n]{n} - 1}n$$

I have tried ratio test/ root test and tried to compare with other convergent series but I do not get anywhere. Also, limit comparison of $a_n$ didn't seems to work.

Since $\lim_{x\to 0^+}x\ln x= 0$ we have  $\exp(x\ln x) \sim 1+x\ln x$, and hence,   $$\frac{n^{1/n} - 1}{n} = \frac1n\left(\exp\left(-\frac1n\ln(\frac1n)\right)-1\right)\sim - \frac1{n^2}\ln(\frac1n) = \frac{\ln n}{n^2}$$
hence the convergence follows by Bertrand criteria or by Cauchy condensation test. or comparison test as follows 
$$n^{3/2}\frac{\ln n}{n^2}= \frac{\ln n}{n^{1/2}}\to 0,~~as~~n\to\infty\implies \frac{\ln n}{n^2}\le \frac{C}{n^{3/2}}$$
