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Thread: Help with series convergence or divergence

  1. #1
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    Help with series convergence or divergence

    I need help to determine if

    $\displaystyle \frac{ln n}{n}$

    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 $\displaystyle \frac{ln n}{n}$ 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 $\displaystyle \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 $\displaystyle \sum_{n=1}^{\infty}\frac{\ln(n)}{n}$ you can do a couple things.

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

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

    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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