The sum to infinity of the series
`1+2(1-(1)/(n))+3(1-(1)/(n))^(2)+ . . .. . . .`, is (A)`n^(2)` (B)`n(n+1)` (C)`n(1+(1)/(n))^(2)` (D)None of these

Options

A. `n^(2)`
B. n(n+1)
C. `n(1+(1)/(n))^(2)`
D. none of these

Answer

A

Explanation

Let `x=1-(1)/(n)` and S be the required sum. Then
`S=1+2x+3x^(2)+4x^(2)+ . . .`
`rArr" "Sx=x+2x^(2)+3x^(3)+ . . .`
`rArr" "S(1-x)=1+x+x^(2)+x^(3)+ . . .`
`rArr" "S(1-x)=(1)/(1-x)`
`rArr" "S=(1)/((1-x)^(2))rArrS=(1)/([1-(1-1//n)]^(2))=n^(2)`