Question

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

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

Answer 1

The correct option is C $n^2$

The given series is in airthmetico geometric progression
the sum of infinite terms of this progression is given as
sum $S_n=a\frac{b}{1-r}+\frac{dbr}{(1-r{)}^2}$
$a=1,d=1,r=(1-\frac{1}{n}),b=1$
Therefore, $S_{Iinfi}=1\frac{1}{1-(1-\frac{1}{n})}+\frac{1\times 1\times (1-\frac{1}{n})}{(1-(1-\frac{1}{n}){)}^2}$
$=n+n(n^2-1)=n^2$