I’m trying to find out if the series

$$sum_{n=2}^infty frac{1}{nn^{1/n}}$$

converges or not. First with the ratio test and then with the integral test.

Ratio test:

$$r=lim_{nrightarrow infty}frac{a_{n+1}}{a_n}=limfrac{nn^{1/n}}{(1+n)(1+n)^{1/(n+1)}}$$

$$lim_{nrightarrow infty}frac{n}{n+1}=1$$

$$lim_{nrightarrow infty}n^{1/n}=lim e^{ln(n)/n}=1$$

$$lim_{nrightarrow infty}(n+1)^{1/(n+1)}=lim e^{ln(n+1)/(n+1)}=e^0=1$$

That means $r=1$ so ratio text is inconclusive.

Integral test

$$int^L frac{1}{xx^{1/x}}dx=int^Lfrac{1}{xe^{ln(x)/x}}dx$$

Let $ln(x)=y$ then

$$intfrac{1}{expleft{ye^{-y}right}}dy $$

I don’t know, what do now? How to show if this integral will or will not converge as $Lrightarrow infty$?