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Math Help - Convergence of a sequence

  1. #1
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    Convergence of a sequence

    Does the following converge?

    \sum_{n=1}^{\infty}\frac{\log n}{n^2}

    Thanks in advance,
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  2. #2
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    Recall that:

    1) \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^p} converges for p > 1
    2) \displaystyle \lim_{n \to \infty} \frac{logn}{n^{\epsilon}} = 0 for every  \epsilon > 0

    And use the comparison test.
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  3. #3
    MHF Contributor chisigma's Avatar
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    The so called 'Riemann zeta function' is defined as...

    \displaystyle \zeta (s) = \sum_{n=1}^{\infty} \frac{1}{n^{s}} , \Re (s) > 1 (1)

    In You derive the series (1) 'term by term' You obtain...

    \displaystyle \zeta^{'} (s) = - \sum_{n=1}^{\infty} \frac{\ln n }{n^{s}} (2)

    ... so that is...

    \displaystyle \sum_{n=1}^{\infty} \frac{\ln n}{n^{2}} = - \zeta^{'} (2) (3)

    Kind regards

    \chi \sigma
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