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Math Help - Can you sum this series analytically?

  1. #1
    Junior Member NowIsForever's Avatar
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    Can you sum this series analytically?

    \sum^{\infty}_{n = 0}(2n+1).4^{n}

    It's in reference to this (un)real world problem: A rat is put in a box with two exit doors.? - Yahoo! Answers

    Thanks for any help!
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by NowIsForever View Post
    \sum^{\infty}_{n = 0}(2n+1).4^{n}

    It's in reference to this (un)real world problem: A rat is put in a box with two exit doors.? - Yahoo! Answers

    Thanks for any help!
    Consider the series

    \displaystyle \sum^{\infty}_{n = 0}(2n+1)x^{n}\bigg|_{x=.4} = 2\sum_{n=0}^{\infty}nx^n+\sum_{n=0}^{\infty}x^n=2x  \sum_{n=0}^{\infty}\frac{d}{dx}x^n+\sum_{n=0}^{\in  fty}x^n

    \displaystyle 2x\frac{d}{dx}\left( \frac{1}{1-x}\right)+\frac{1}{1-x}=\frac{2x}{(1-x)^2}+\frac{1}{1-x}=\frac{1+x}{(1-x)^2}

    Now just plug in x=0.4

    This gives

    \displaystyle \frac{1.4}{(.6)^2}=3.\bar{8}
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  3. #3
    Junior Member NowIsForever's Avatar
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    I wrote the wrong problem. It should have been

    \displaystyle \sum^{\infty}_{n = 0}(2n+3)x^{n}=\frac{3-x}{(1-x)^2}

    Which is 65/9 when x = .4, and is in accord with the WolframAlpha result.

    Thanks!
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