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How do i prove that the following series diverges?

  1. Apr 13, 2009 #1
    1/ln((n)!) [n goes from 2-infinity]

    1/ln((2)!) + 1/ln((3)!) +1 /ln((4)!) +......+ 1/ln((n)!)

    the first thing i thought of was de lambert

    lim An+1/An
    n->inf

    =ln(x!)/ln((x+1)!)
    =ln(x!)/ln(x!*(x+1))
    =ln(x!)/[ln(x!)+ln(x+1)]

    =1-ln(x+1)/[ln(x!)+ln(x+1)]=1
    which doesnt tell me anything about the convergence /divegence of the series.

    have i done something wrong somewhere here, or is there another way to solve this?
     
  2. jcsd
  3. Apr 14, 2009 #2

    CompuChip

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    Science Advisor
    Homework Helper

    I don't know if this is the most efficient way, but if you note that
    ln(n!) = ln(n) + ln(n - 1) + ... <= n ln(n)
    you can show that
    [tex]S = \sum_{n = 2}^\infty \frac{1}{\ln(n!)} \ge \sum_{n = 2}^\infty \frac{1}{n \ln n}[/tex]
     
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