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Math Help - Cauchy condensation test

  1. #1
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    Cauchy condensation test

    I need to determine, using the Cauchy Condensation Test, whether or not
    the series 1/(n * Log(n)) converges. I believe that this series converges iff
    2^n(1/(2^n*Log(2^n)) converges and I believe that it actually diverges. But I am not sure how to work through it. Thanks for your help.
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  2. #2
    Senior Member roninpro's Avatar
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    It looks like you already have it! You wrote down \displaystyle \sum_{n=1}^\infty 2^n\frac{1}{2^n \log (2^n)}. This is precisely \displaystyle \sum_{n=1}^\infty \frac{1}{\log (2^n)}=\sum_{n=1}^\infty \frac{1}{n \log 2}=\displaystyle \frac{1}{\log 2}\sum_{n=1}^\infty \frac{1}{n}, which diverges.
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