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Proving Convergence of this

  1. Mar 22, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex]\sum[/tex] from n=2 to [tex]\infty[/tex] of n/((n2-5)*(ln n)2)

    2. Relevant equations



    3. The attempt at a solution
    I've tried Limit Comparison but I always get a limit of 0 which will not work. Ratio test doesn't help. I don't think a direct comparison can be made but that seems to be the only other option...
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 22, 2009 #2
    Hi simba924!

    To simplify the problem a bit, show that

    [tex]\frac{n}{n^2-5}\le\frac{2}{n}[/tex]

    for large n. Then apply the http://en.wikipedia.org/wiki/Integral_test_for_convergence" [Broken].
     
    Last edited by a moderator: May 4, 2017
  4. Mar 22, 2009 #3
    Ahh thanks. Can you explain that in a little more detail though, I don't really get it.
     
  5. Mar 22, 2009 #4
    Hey I still need some help with this one. I'm pretty sure that [tex]\sum[/tex] 2/n is not convergent because it is in form integer/np where p=1

    Can anyone help?
     
  6. Mar 23, 2009 #5
    Remember, you still have the additional factor 1/(ln n)^2 (without which it would diverge). To do the integral test you have to check that

    [tex]\int_2^{\infty}\frac{1}{x(\log(x))^2}[/tex]

    exists (you can find an explicit antiderivative, Hint: substitution).
     
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