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Math Help - Find the sum of the serie explicitly

  1. #1
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    Find the sum of the serie explicitly

    |\omega|\leq 1, \omega\neq1 and \omega is complex
    \Sigma_{n=0}^\infty \frac{\omega^n}{n}

    Find the sum of this serie explicitly



    Any ideas would be appreciated, thanks
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by KZA459 View Post
    |\omega|\leq 1, \omega\neq1 and \omega is complex
    \Sigma_{n=0}^\infty \frac{\omega^n}{n}

    Find the sum of this serie explicitly



    Any ideas would be appreciated, thanks
    First I really hope that n=1 is where it starts or we have a problem...

    With that said notice that if

    f(\omega)=\sum_{n=1}^\infty \frac{\omega^n}{n}

    we can take the derivative both sides to get

    f'(\omega)=\sum_{n=1}^\infty \omega^{n-1}

    If we reindex the series we get

    f'(\omega)=\sum_{n=1}^\infty \omega^{n-1}=\sum_{n=0}^{\infty}w^{n}=\frac{1}{1-\omega}

    Just integrate from here and you are done
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  3. #3
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    Thanks a lot and yes it was from 1 to infinity
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