Find the sum `(1^2)/(2)+(3^2)/(2^2)+(5^2)/(2^3)+(7^2)/(2^4)+….oo`

Answer

17

Explanation

`S=1^(2)/2+3^(2)/2^(2)+5^(2)/2^(3)+7^(2)/2^(4)+….oo` (1)
`rArr1/2s=1^(2)/2^(2)+3^(2)/2^(3)+5^(2)/2^(4)+…oo` (2)
`rArr1/2S=1^(2)/2^(2)+3^(2)/2^(3)+5^(2)/2^(4)+…oo` (2)
Subtracting (2) from (1),
`1/2S=1^(2)/2+8/2^(2)+16/2^(3)+24/2^(4)+32/2^(5)+..oo` (3)
Let `S_(1)=8/2^(2)+16/2^(3)+24/2^(4)+32/2^(5)+..oo` (4)
`rArr1/2S_(1)=8/2^(3)+16/2^(4)+24/2^(5)+...oo` (5)
Subtracting (5) from (4),
`1/2S_(1)=8/2^(2)+8/2^(3)+8/2^(4)+8/2^(5)+...oo`
Subtracting (5) from (4),
`1/2S_(1)=8/2^(2)+8/2^(3)+8/2^(4)+8/2^(5)+...oo`
`=(8/2^(2))/(1-1/2)`
=4
or `S_(1)=8`
Hence, from (3),
`1/2S=1/2+8`
`rArrS=17`