Convergence of the $\sum_{1}^\infty n^{\frac1n}$

Question

I stumbled upon that question tried some tests like ratio test and ahare test but all gave the limit 1 which is indecisive. Any one with a better approach..

Answer 1

$n^{\frac{1}{n}} = e^{\frac{\log n}{n}} \xrightarrow[n\to\infty]{} e^0= 1$, so the term of the series does not converges to $0$ and the series diverges to $\infty$ (necessary condition for convergence of a series: general term going to $0$).