Question

This sum to infinity of the series, $1+2(1-\frac{1}{n})+3{(1-\frac{1}{n})}^2+...$ is

• A. $n^2$
• B. $n(n+1)$
• C. $n{(1+\frac{1}{n})}^2$
• D. None of these.

Answer 1

Answer: (A)

$n^2$

$S_{\infty }=1+2[1-\frac{1}{n}]+3{[1-\frac{1}{n}]}^2+.....\infty$
$[1-\frac{1}{n}]S_{\infty }=[1-\frac{1}{n}]+2{[1-\frac{1}{n}]}^2+3{[1-\frac{1}{n}]}^3+.....\infty$
$S_{\infty }-[1-\frac{1}{n}]S_{\infty }=1+[1-\frac{1}{n}]+{[1-\frac{1}{n}]}^2+{[1-\frac{1}{n}]}^3+.....\infty$
$\frac{n{S}_{\infty }-n{S}_{\infty }+S_{\infty }}{n}=\frac{1}{1-[1-\frac{1}{n}]}$ ..... $[\because \text{Sum of infinite G.P.}=\frac{a}{1-r}]$
$\frac{S_{\infty }}{n}=\frac{n}{n-n+1}$
$S_{\infty }=n^2$