Question

The value of sum $\sum _{n=1}^{\infty }\frac{n}{7^n}$ is

• A. $\frac{6}{7}$
• B. $\frac{7}{6}$
• C. $\frac{49}{36}$
• D. $\frac{7}{36}$

Answer 1

The correct option is D $\frac{7}{36}$

Let $S=\sum _{n=1}^{\infty }\frac{n}{7^n}$
$\Rightarrow S=\frac{1}{7}+\frac{2}{7^2}+\frac{3}{7^3}+\frac{4}{7^4}+\cdots \Rightarrow \frac{S}{7}=\frac{1}{7^2}+\frac{2}{7^3}+\frac{3}{7^4}+\frac{4}{7^5}+\cdots \Rightarrow \frac{6S}{7}=\frac{1}{7}+\frac{1}{7^2}+\frac{1}{7^3}+\frac{1}{7^4}+\cdots \Rightarrow \frac{6S}{7}=\frac{\frac{1}{7}}{1-\frac{1}{7}}\therefore S=\frac{7}{36}$