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N-th root of n factorial

  1. Sep 19, 2007 #1
    I'm working on the limit of the sequence
    (Xn) = (n!)^(1/n)
    Pretty sure it diverges as n goes to infinity,
    but unsure how to show it.
    Any hint would ge greatly appreciated.
  2. jcsd
  3. Sep 19, 2007 #2


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    Your instincts are good. Take a log and use Stirling's formula. That's one way.
  4. Sep 19, 2007 #3
    an elemetary proof is to take log so you get (1/n)log[n]! then you get [log1+Log2+log3...logn]/n

    set bounds for 1<n<11 sum truncate terms
    you get sum between 0 and 1

    for 11<n<101 get sum between 10 and 20

    you can see how this diverges
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