Question

$\frac{3}{1^2{.2}^2}+\frac{5}{2^2{.3}^2}+\frac{7}{3^2{.4}^2}$+.....n terms and deduce the sum upto infinity.

Answer 1

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

Putting n = 1, 2 , 3.........n and adding we get

$S_n=-1\frac{1}{(n+1{)}^2}=\frac{n^2+2n}{n^2+2n+1}{s}_{\infty }=L{t}_{n\to \infty }\frac{n^2(1+2/n)}{n^2(1+2/n+1/n^2)}=1$