Find the sum to `n` terms of the series: `3/(1^2 .2^2)+5/(2^2 .3^2)+7/(3^2 .4^2)+......`

Options

A. 0
B. 2
C. `(1)/(2)`
D. 1

Answer

D

Explanation

Let `T_(r)" be the "r^(th)` term of the given series. Then,
`T_(r)=((2r+1))/(r^(2)(r+1)),r=1,2,3, . . . . . `
`rArr" "T_(r)={(1)/(r^(2))-(1)/((r+1)^(2))},r=1,2,3, . . . `
Let S be the required sum. Then,
`S=underset(nto oo)limunderset(r=1)overset(n)sumT_(r)`
`rArr" "S=underset(nto oo)limunderset(r=1)overset(n)sum{(1)/(r^(2))-(1)/((r+1)^(2))}=underset(nto oo)lim{1-(1)/((n+1)^(2))}=1`