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Evaluate the sum \(\frac{1}{3^1} + \frac{2}{3^2} + \frac{3}{3^3} + \cdots + \frac{k}{3^k} + \cdots \)

 Aug 15, 2019
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\(S = \sum \limits_{k=0}^\infty ~x^k= \dfrac{1}{1-x}\\ \dfrac{dS}{dx} = \sum \limits_{k=1}^\infty k x^{k-1} = \dfrac{1}{(1-x)^2}\\ \sum \limits_{k=1}^\infty k x^k = \dfrac{x}{(1-x)^2}\\~\\ \sum \limits_{k=1}^\infty ~\dfrac{k}{3^k} = \dfrac{\frac 1 3}{\left(1-\frac 1 3\right)^2}= \dfrac 3 4\)

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 Aug 15, 2019
edited by Rom  Aug 15, 2019

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