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Math Help - Converges or Diverges

  1. #1
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    Converges or Diverges

    \sum_{k=2}^\infty \frac{4}{k ln k}

    should I turn the ln into an e? What rule should I use??
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by kl.twilleger View Post
    \sum_{k=2}^\infty \frac{4}{k ln k}

    should I turn the ln into an e? What rule should I use??
    Let a_k=\frac{4}{k\ln(k)}. Consider that \forall{k}\in[2,\infty)~a_k>0\wedge{a_k}\in\mathcal{C}\wedge{a_k  }\in\downarrow so the integral test applies. So this series shares convergence/divergence with \int_2^{\infty}\frac{dx}{x\ln(x)}
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  3. #3
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    There is an alternative solution to this.

    Theorem. If a_1 > a_2 > ... > 0 then series \sum_{k = 1}^\infty a_k converges if and only if \sum_{k = 1}^\infty 2^k a_{2^k} converges.
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by albi View Post
    There is an alternative solution to this.

    Theorem. If a_1 > a_2 > ... > 0 then series \sum_{k = 1}^\infty a_k converges if and only if \sum_{k = 1}^\infty 2^k a_{2^k} converges.
    Note to the poster, this is known as Cauchy's Condensation test and is rarely taught at a Calculus level course. It is useful in rare cases, but when it is applicable it is usually powerful.
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