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Math Help - trying to find if a particular series converges or diverges

  1. #1
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    trying to find if a particular series converges or diverges

    Hey all,

    I have the following series \sum_{n=1}^{\infty}(\frac{ln(n)}{n})^2 . I used the integral test, but since the function has a maximum at e not sure if its the right test. I got 2/e which is < 1 hence the series converges.

    Thanks, for all thelp..
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: trying to find if a particular series converges or diverges

    Denote u_n=\left(\dfrac{\ln n}{n}\right)^2 and v_n=n^{3/2} then, \frac{u_n}{v_n}=\frac{\ln ^2 n}{n^{1/2}}\to 0 as n\to +\infty . Being \sum_{n=1}^{+\infty}v_n convergent, also \sum_{n=1}^{+\infty}u_n is convergent.

    Remark: This is a particular case of the Bertrand series.
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  3. #3
    MHF Contributor
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    Re: trying to find if a particular series converges or diverges

    Quote Originally Posted by Oiler View Post
    Hey all,

    I have the following series \sum_{n=1}^{\infty}(\frac{ln(n)}{n})^2 . I used the integral test, but since the function has a maximum at e not sure if its the right test. I got 2/e which is < 1 hence the series converges.

    Thanks, for all thelp..
    You can apply the integral test if you start your series at n = 3 so \sum_{n=3}^{\infty}(\frac{ln(n)}{n})^2 .
    Adding a few terms (or even subtracting a few) won't change the converence of the series.
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  4. #4
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    Re: trying to find if a particular series converges or diverges

    Echoing Danny's comments, the more general requirement for the intergral test is that there must exist some N such that, for all n \ge N, a_n >0. That is, it only matters that the "tail" of the sequence is positive.
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