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Math Help - Help with series convergence or divergence

  1. #1
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    Help with series convergence or divergence

    I need help to determine if

    \frac{ln n}{n}

    converges or diverges. By the nth term test it equals 0 so it could be either. I remember doing this in class but I forgot.
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  2. #2
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    Quote Originally Posted by Cakecake View Post
    I need help to determine if \frac{ln n}{n} converges or diverges. By the nth term test it equals 0 so it could be either. I remember doing this in class but I forgot.
    Do you mean \sum\limits_{n = 1}^\infty  {\frac{{\ln (n)}}{n}} ?
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  3. #3
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    Ah yes, I forgot the summation.
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  4. #4
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    To determine the convergence of \sum_{n=1}^{\infty}\frac{\ln(n)}{n} you can do a couple things.

    1) Use the integral test which states that if a_n is continuous, nonincreasing, and positive that \sum_{n=c}^{\infty} a_n and \int_c^{\infty} f~dx share convergence

    2) Use Cauchy's condensation test. Once again if the same hypotheses apply then \sum a_n and \sum a_{2^n}\cdot 2^n share convergence.
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  5. #5
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    Quote Originally Posted by Plato View Post
    Do you mean \sum\limits_{n = 1}^\infty {\frac{{\ln (n)}}{n}} ?
    Note that \frac{\ln (n)}{n} > \frac{1}{n} for n > 2.

    So use the comparison test to prove divergence ....
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  6. #6
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    Ooo right thanks, that's what I did over in class. Didn't bother thinking of it because I was using that to compare another series.
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