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Determining convergence of (-1)^n/ln(n!) as n goes to infinity

  1. Nov 26, 2011 #1
    1. The problem statement, all variables and given/known data

    Determine whether (-1)^n/ln(n!) is divergent, conditionally convergent, or absolutely convergent.



    2. Relevant equations
    None, really? :S


    3. The attempt at a solution

    Okay, so I know the series converges by the Alternating Series test - terms are positive, decreasing, going to zero. I also know that it absolutely converges...because Wolfram Alpha told me so. =P I have no idea how to prove it though. The ratio test results in 1 so that's inconclusive, so I'm left with comparison or limit comparison test..but I don't know what to compare it to.

    Thanks in advance for your help!
     
  2. jcsd
  3. Nov 26, 2011 #2

    Dick

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    I would compare it to 1/ln(n^n). That's less than 1/ln(n!), right? And 1/ln(n^n) diverges by an integral test. Do you have that?
     
  4. Nov 26, 2011 #3
    WolframAlpha tells you the answer, you have the solutions. I don't see what's wrong here
     
  5. Nov 26, 2011 #4

    Dick

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    Why are you posting here if you have nothing to contribute??
     
    Last edited: Nov 27, 2011
  6. Nov 26, 2011 #5
    Hi Dick and FlyingPig,

    Thanks for your responses.

    Dick - you are right, the integral does not converge.

    FlyingPig, the problem is that WolframAlpha gave me an integer answer when I entered "the summation of 1/ln(n!) from 2 to infinity", so that was misleading.

    Anyway thanks so much for helping me solve the problem.
     
  7. Nov 26, 2011 #6
    This is what WolframAlpha gave me:

    NIntegrate::ncvb: "NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in n near {n} = {2.176*10^6}. NIntegrate obtained 2.1848934935880644` and 0.0050671018999950595` for the integral and error estimates."

    and then the integer 5.49193 as output. I guess that was an estimate for a sum that never actually converges.
     
  8. Nov 26, 2011 #7

    Dick

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    You are quite right. Wolfram Alpha can be misleading. So can flyingpig. Can you show the series diverges without trusting either one? What's the integral of 1/(n*ln(n))?
     
    Last edited: Nov 26, 2011
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