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Math Help - Sum of a sequence

  1. #1
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    Sum of a sequence

    \sum_{x=0}^{\infty}\frac{x}{2^x}

    All I know is that it converges. Really have nothing more to say.
    Last edited by lausing; February 21st 2010 at 09:51 PM. Reason: latex
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  2. #2
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    Hello, lausing!

    S \;=\;\sum^{\infty}_{x=1} \frac{x}{2^x}

    \begin{array}{cccccc}\text{We have:} & S &=& \dfrac{1}{2} + \dfrac{2}{2^2} + \dfrac{3}{2^3} + \dfrac{4}{2^4} + \hdots \\ \\[-3mm]<br />
\text{Multiply by }\frac{1}{2}\!: & \frac{1}{2}S &=& \quad\;\;\dfrac{1}{2^2} + \dfrac{2}{2^3} + \dfrac{3}{2^4} + \hdots \end{array}


    . . \text{Subtract: }\;\;\tfrac{1}{2}S \;\;=\;\;\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + \hdots .[1]


    The right side is a geometric series with: . a = \tfrac{1}{2},\;\;r = \tfrac{1}{2}

    . . Its sum is: . \frac{\frac{1}{2}}{1-\frac{1}{2}} \:=\:1


    Hence [1] becomes: . \tfrac{1}{2}S \;=\;1


    Therefore: . S \;=\;2

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  3. #3
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    Thanks!

    Also, the question is erroneously posted in calculus rather than linear algebra or something, since it orginally was about derivatives. Whoops!
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