Summing n plus one over two to the n

Evaluate $\displaystyle\sum_{n=1}^\infty \dfrac{n+1}{2^{n}}$ . (to 3 decimals)

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Split $\tfrac{n+1}{2^n}$ into $\tfrac{n}{2^n} + \tfrac{1}{2^n}$ .
For $\sum n x^n$ , differentiate $\tfrac{1}{1-x}$ and multiply by $x$ ; here $x=\tfrac12$ .

Answer

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3 (± 0.001)

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Asked at: Probability & Market-Making, Options Market-Making