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Lim n to infinity for factorial

  1. Sep 10, 2008 #1
    1. The problem statement, all variables and given/known data
    lim n -> infinity for (n!)^(1/n)


    2. Relevant equations



    3. The attempt at a solution
    hmm, i know that lim n approaches infinity, (n)^(1/n) will go to 1, but issit the same for n!?
     
  2. jcsd
  3. Sep 10, 2008 #2

    HallsofIvy

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    You might do this: n!= 1*2*3*...* (n-1)*n so it has exactly n factors. (n!)^(1/n)= (1)^(1/n)(2)^(1/n)(3)^(1/2)*...*(n-1)^(1/n)*n^(1/n). Now you say that you know that n^(1/n) goes to 1. What do you think the other numbers go to? In particular, what does 2^(1/n) or 3^(1/n) go to? If you don't know try looking at 2^(1/100000) or 3^(1/100000). What does the product of thing like that go to?
     
  4. Sep 10, 2008 #3

    Dick

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    ?? All of those limits may be one. But the limit of the product certainly isn't one. Use Stirling's formula for n!.
     
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