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If S= 13+232+...

Question

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

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Solution

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

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

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