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Math Help - Integral Test

  1. #1
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    Integral Test

    Hi, im trying to perform an integral test on the following:

    \sum _{n=1}^{\infty }{\frac {1}{n\ln  \left( n+1 \right) }}

    I know the theory is to integrate the function and then to evaluate it and see if it is finite... however i am having some problems with the integration. Can anyone help?
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  2. #2
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    Quote Originally Posted by Mathman87 View Post
    Hi, im trying to perform an integral test on the following:

    \sum _{n=1}^{\infty }{\frac {1}{n\ln  \left( n+1 \right) }}

    I know the theory is to integrate the function and then to evaluate it and see if it is finite... however i am having some problems with the integration. Can anyone help?
    use the integral test to show \sum_{n=2}^\infty \frac{1}{n\ln{n}} diverges ... then use the limit comparison test with this divergent series to show \sum_{n=1}^\infty \frac{1}{n\ln(n+1)} also diverges.
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  3. #3
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     \sum _{n=1}^{\infty }{\frac {1}{n\ln \left( n+1 \right) }} behaves like  \sum _{n=1}^{\infty }{\frac {1}{n\ln \left( n \right) }} ; thus,  \int_1^\infty \frac{1}{n\ln(n)} dn = \left(\ln \ln(n)\right) |_1^\infty
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