Test for convergence, a sequence with $n^{th}$ term $a_n=n(\sqrt[\phantom{2}]{n^2+1}-n)$

Test for convergence: ${\displaystyle \underset{n=1}{\overset{}{}}}ta{n}^{-1}(1-\frac{2}{n})$

Test for convergence: ${\displaystyle \underset{n=2}{\overset{}{}}}\frac{cosn}{(lnn{)}^2}$