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Math Help - convergence of series.

  1. #1
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    convergence of series.

    How does one prove that the series

    1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + 1/9 + 1/11 - 1/6 + ...

    converge?

    PS.: This series is a rearrangement of the alternating harmonic series in which two positive terms are always followed by one negative.
    Last edited by mr fantastic; November 26th 2010 at 04:33 AM. Reason: Title.
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  2. #2
    MHF Contributor chisigma's Avatar
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    The series converges... but not as \displaystyle 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ... = \ln 2 because that is a weakly convergent series...

    Kind regards

    \chi \sigma
    Last edited by chisigma; November 25th 2010 at 10:59 PM. Reason: ortographic error...
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  3. #3
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    Quote Originally Posted by jefferson_lc View Post
    How does one prove that the series

    1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + 1/9 + 1/11 - 1/6 + ...

    converge?

    PS.: This series is a rearrangement of the alternating harmonic series in which two positive terms are always followed by one negative.

    Its limit is \displaystyle{\frac{3}{2}\ln 2} .

    The proof appears in Bonar-Khouri's "Real Infinite Series", in 3.10.

    Tonio
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