$an>1,0\frac{si{n}^2(n)}{n^2}\frac{1}{n^2},$ and the series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{n^2}$ converges, $soby$ the Comparison Test, the series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{si{n}^2(n)}{n^2}$ converges.

and the series $\frac{}{2}{\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{n^3}$ converges, $soby$ the Comparison Test, the series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{arctan(n)}{n^3}$

converges.

$cn>2,\frac{log(n)}{n^2}>\frac{1}{n^2}0,$ and the series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{n^2}$ converges, $soby$ the Comparison Test, the series ${\displaystyle \underset{n=2}{\overset{}{}}}\frac{log(n)}{n^2}$ converges.

and the series $2{\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{n^2}$ converges, $soby$ the Comparison Test, the series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{n}{n^3-6}$ converges.

and the series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{n^2}$ converges, $soby$ the Comparison Test, the series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{n^2-8}$ converges.

and the series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{n^2}$ converges, $soby$ the Comparison Test, the series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{n}{2-n^3}$ converges.