Question

Let $r=\frac{1}{2}$, consider $\sum _{n=0}^{\infty }n{r}^n$ for increasing value of $n$ and then find its value as $n$ tends to infinity.

• A. Tends to $1$
• B. Tends to $0$
• C. Tends to $2$
• D. None of the above

Answer 1

Answer: (C)

Tends to $2$

Given:

$r=\frac{1}{2}$

Let $S=\sum _{n=0}^{\infty }n{r}^n$ ......(1)

$\therefore rS=\sum _{n=0}^{\infty }n{r}^{n+1}$ .....(2)

eq (1)- eq (2)

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

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

Hence, $\sum _{n=0}^{\infty }n{r}^n$ tends to $2$.