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Math Help - find the sum of the series

  1. #1
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    find the sum of the series

    Find the sum of the series \sum_{n = 1}^\infty \frac{{-1}^{n }*{4}^{n }  }{ n! n}
    Last edited by topsquark; May 12th 2011 at 05:13 PM. Reason: Fixed LaTeX code
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by jayantamath View Post
    Find the sum of the series \sum_{n = 1}^\infty \frac{{-1}^{n }*{4}^{n }  }{ n! n}

    If you mean

    \sum_{n=1}^{+\infty}\dfrac{(-1)^n4^n}{n!n}

    then, consider the functional series

    f(x)=\sum_{n=1}^{+\infty}\dfrac{(-1)^nx^n}{n!n}

    f'(x)=\dfrac{1}{x}\sum_1^{+\infty}\dfrac{(-1)^nx^n}{n!}=\dfrac{e^{-x}-1}{x}

    You'll obtain

    f(4)=\textrm{Ei}(-4)-\gamma-\log 4

    where

    \textrm{Ei}(x)=-\int_{-x}^{+\infty}\dfrac{e^{-t}dt}{t}

    and \gamma the Euler-Mascheroni constant.
    Last edited by FernandoRevilla; May 13th 2011 at 01:31 AM.
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