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This question already has an answer here:

the sum of the series $\sum_{n=1}^\infty \frac{1}{2^{n}}$ is 1. It is easy to find since it is a g.p. the series $\sum_{n=1}^\infty \frac{n}{2^{n}}$ is convergent by ratio test. How will find the infinite sum? I am trying to rearrange the terms and to use the rearrangement theorem, but I can't complete

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marked as duplicate by vadim123, lab bhattacharjee, Hans Lundmark, Community Mar 5 '18 at 16:18

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HINT: Use that $\sum_{n=0} nx^{n-1}=(\sum_{n=0} x^n)'=(\frac{1}{1-x})'$ for $x=\frac{1}{2}$.

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