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Math Help - Cauchy Condensation Test

  1. #1
    Member disclaimer's Avatar
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    Cauchy Condensation Test

    Hi;

    Can somebody please provide me with an example of using the Cauchy condensation test (criterion " 2^k") in checking convergence of series? I've never used it myself, and I'd really like to see a real example of application (it seems rather rare to use this criterion), not just the rule. I'd like to know how it works in practice.

    Thanks for your time.
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  2. #2
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    Quote Originally Posted by disclaimer View Post
    Hi;

    Can somebody please provide me with an example of using the Cauchy condensation test (criterion " 2^k") in checking convergence of series? I've never used it myself, and I'd really like to see a real example of application (it seems rather rare to use this criterion), not just the rule. I'd like to know how it works in practice.

    Thanks for your time.
    Have you read this?
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  3. #3
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    I suppose that you already know that \sum^\infty_{k=1}a_k is a convergent series iff \sum^\infty_{k=1}2^ka_{2k} is a convergent series. The Cauchy Condensation test can mainly be used to prove divergence for various series and we can firstly use this to prove the divergence of the harmonic series

    Notice how
    \sum_{k=1}^\infty\frac{1}{k} transforms to \sum^\infty_{k=1}\frac{2^k}{2^k}=\sum_{k=1}^\infty  1
    under the Cauchy condensation test and hence diverges

    I very much hope that that helps,

    Paul
    Last edited by notyeteuler; March 6th 2008 at 04:34 AM.
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