The sum of the series `1+2(1 +1/n)+ 3(1+1/n)^2+.... oo` is given by

Options

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

Answer

A

Explanation

Let S be the sum of n terms of the given series and `a=1+1//n`. Then,
`S=1+2x+3x^(2)+4x^(2)+ . . . +nx^(n-1)`
`rArr" "xS=x+2x^(2)+3x^(3)+ . . . +(n-1)x^(n-1)+nx^(n)`
`:." "S-xS=1+{x+x^(2)+ . . . +x^(n-1)}-nx^(n)`
`rArr" "S(1-x)=(1-x^(n))/(1-x)-nx^(n)`
`rArr" "S(-1//n)=-n[1-(1+1//n)^(n)]-n(1+1//n)^(2)`
`rArr" "(1)/(n)S=n{(1-(1+1//n)^(n)+(1+1//n)^(n)}=n`
`rArr" "S=n^(2)`