What is the sum of the series
`1+(1)/(8)+(1.3)/(8.16)+(1.3.5)/(8.16.24)+…oo ?`

Options

A. `(2)/(sqrt(3))`
B. `2sqrt(3)`
C. `(sqrt(3))/(2)`
D. `(1)/(2sqrt(3))`

Answer

A

Explanation

As given the series is
`S=1+(1)/(8)+(1.3)/(8.16)+(1.3.5)/(8.16.24)+...oo`
On comparing this series with
`S=(1+x)^(n)=1+nx+(n(n-1))/(2!)x^(2)+…oo,` we get
`nx=(1)/(8)" "...(1)`
and `(n(n-1))/(2!)x^(2)=(1.3)/(8.16)" "...(2)`
From Eqs. (1) and (2), we get
`((n(n-1))/(2!)x^(2))/(n^(2)x^(2))=((1.3)/(8.16))/((1)/(8).(1)/(8))`
`=(n-1)/(2n)=(3)/(2)`
`implies" "n-1=3n`
`implies" "n=-(1)/(2)`
On putting this value in Eq. (i)
`implies" "(-(1)/(2))x=(1)/(8)`
`implies" "x=-(1)/(4).`
But `S=(1+x)^(n)=(1-(1)/(4))^(-1//2)`
`=((3)/(4))^(-1//2)=(2)/(sqrt(3)).`