1st)I need to investigate convergence usingconvergence of integral feature

$\displaystyle \sum_{n=1}^{inf}\frac{1}{\sqrt{n}}$

From limit I see that it's 0, but how do I prove it using that method?

2nd)Here I need to investigate convergence usingseries required condition of convergence

$\displaystyle \sum_{n=1}^{\inf}\cos^2n$