Answer : (a) $\;\large\frac{n(n+2)}{(n+1)^2}$
Explanation : $a_{r}=\large\frac{2r+1}{r^2.(r+1)^2}$
$=\large\frac{(r+1)^2-r^2}{r^2\;.(r+1)^2}$
$=\large\frac{1}{r^2}-\large\frac{1}{(r+1)^2}$
$S_{n}=\sum_{r=1}^{n}\;a_{r}$
$=\sum_{r=1}^{n}\;\large\frac{1}{r^2}-\large\frac{1}{(r+1)^2}$
$S_{n}=1-\large\frac{1}{(n+1)^2}$
$=\large\frac{(n+1)^2-1}{(n+1)^2}$
$S_{n}=\large\frac{n(n+2)}{(n+1)^2}\;.$