Question

Sum to $n$ terms the following series
$\frac{3}{1^2\cdot 2^2}+\frac{5}{2^2\cdot 3^2}+\frac{7}{3^2\cdot 4^2}+......$

Answer 1

$S=\frac{3}{1^2\cdot 2^2}+\frac{5}{2^2\cdot 3^2}+\frac{7}{3^2\cdot 4^2}+$ .......

$=\frac{2^2-1^2}{1^2{.2}^2}+\frac{3^2-2^2}{2^2-3^2}+........\frac{(n+1{)}^2-m^2}{n^2(n+1{)}^2}$

$=(1-\frac{1}{2^2})+(\frac{1}{2^2}-\frac{1}{3^2})+......(\frac{1}{n^2}-\frac{1}{(n+1{)}^2})$

$=1-\frac{1}{(n+1{)}^2}$