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Quick series question

  1. Dec 4, 2007 #1
    why does the series 1/(nln(n)) diverge? I thought it converged since the limit goes to 0.
     
  2. jcsd
  3. Dec 4, 2007 #2
    No, the SEQUENCE

    [tex]\left\{ \frac{1}{n\ln n} \right\}[/tex]

    converges because the limit of the terms go to 0.

    However, the SERIES

    [tex]\sum_{n=2}^\infty \frac{1}{n\ln n}[/tex]

    diverges using the integral test.
     
  4. Dec 4, 2007 #3
    For the series

    [tex]\sum_{n=1}^\infty a_n[/tex]

    the condition that

    [tex]\lim_{n\to\infty} a_n = 0[/tex]

    is necessary for convergence, however it is not sufficient. That is, satisfying the limit condition is not enough to conclude that the series converges.
     
  5. Dec 4, 2007 #4
  6. Dec 4, 2007 #5
    I see, so

    1/n will diverge since p <= 1 and 1/nln(n) is smaller than that, so it will converge as well--is that a correct comparison test?
     
  7. Dec 4, 2007 #6

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    No, it's not. if an converges and bn< an then bn converges. If an diverges and bn> an then bn diverges. If an diverges and bn< an, you don't have any information as to whether bn converges or not.
    As rs1n said, use the integral test.
     
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