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Riemann Sum series problem

  1. Nov 12, 2006 #1

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    For what values of p>0 does the series

    Riemann Sum [n=1 to infinity] 1/ [n(ln n) (ln(ln n))^p]

    converge and for what values does it diverge?

    How do i do this question? Would somebody please kindly show me the steps? Do i use the intergral test?
     
  2. jcsd
  3. Nov 12, 2006 #2

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    Try bounding it by the series 1/n^(1+e), e>0. Note these sums get arbitrarily large as e->0.
     
  4. Nov 13, 2006 #3

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    it is actually a example question, my instructor said that it is divergent for all p, but i don't get it because actually i didn't copy all of the notes. Do you mind explaining it to me?
     
    Last edited: Nov 13, 2006
  5. Nov 13, 2006 #4

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    Oh, actually the integral test does work. You can do it nicely with a substitution. Also, be careful with the first two terms. ln(ln(1))=-infinity and ln(1)=0, so the first term isn't well defined, but even if you take the limit it blows up. And ln(ln(2)) is negative, and so can't be raised to the pth power in a well defined way. I would assume the series is supposed to start at n=3.
     
    Last edited: Nov 13, 2006
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