Question

Prove that the sum to infinity of the series $1+2(1-\frac{1}{n})+3{(1-\frac{1}{n})}^2+...$ where $n\in N$, is given by $n^2$.

Answer 1

Let $S=1+2(1-\frac{1}{n})+3{(1-\frac{1}{n})}^2+...$ ...(1)

$\therefore (1-\frac{1}{n})S=(1-\frac{1}{n})+2{(1-\frac{1}{n})}^2+...$ ...(2)

From (1) and (2)

$\Rightarrow \frac{S}{n}=1+(1-\frac{1}{n})+{(1-\frac{1}{n})}^2+...\infty$

$=\frac{1}{1-(1-\frac{1}{n})}=n\Rightarrow S=n^2$