Question

The value of $\sum _{n=1}^{\infty }\frac{2n}{3^n}$ can be expressed in the form of $\frac{a}{b},$ where $a$ and $b$ are coprime positive integers. Then the value of $a+b$ is

Answer 1

Let $S=\sum _{n=1}^{\infty }\frac{2n}{3^n}=2\sum _{n=1}^{\infty }\frac{n}{3^n}$
$\frac{1}{2}S=\frac{1}{3}+\frac{2}{9}+\frac{3}{27}+\dots$
Then $\frac{1}{2}\cdot \frac{S}{3}=\frac{1}{9}+\frac{2}{27}+\frac{3}{81}+\dots$
$\therefore \frac{1}{2}S-\frac{1}{2}\cdot \frac{S}{3}=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\dots$
$\Rightarrow \frac{S}{3}=\frac{\frac{1}{3}}{1-\frac{1}{3}}=\frac{1}{2}$
$\Rightarrow S=\frac{3}{2}$
Thus, $a+b=5$