Question

If $S=\frac{1}{3}+\frac{2}{3^2}+\frac{3}{3^3}+\frac{4}{3^4}+\cdots \infty ,$ then the value of $4S$ is

Answer 1

$S=\frac{1}{3}+\frac{2}{3^2}+\frac{3}{3^3}+\frac{4}{3^4}+\cdots \infty \frac{S}{3}=\frac{1}{3^2}+\frac{2}{3^3}+\frac{3}{3^4}+\frac{4}{3^5}+\cdots \infty$

Subtracting the above two equations,
$\frac{2S}{3}=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}+\cdots \infty \Rightarrow \frac{2S}{3}=\frac{\frac{1}{3}}{1-\frac{1}{3}}\therefore 4S=3$