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Weird limit…

  1. Aug 2, 2004 #1
    does...

    [tex] \lim_{n \rightarrow \infty} \frac{n}{(n!)^\frac{1}{n}} = e [/tex]

    If not, is it divergent?
     
  2. jcsd
  3. Aug 2, 2004 #2

    mathwonk

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    isnt there somethiong called stirlings formula for n! ?? Maybe you could use that and lhopital.
     
  4. Aug 2, 2004 #3

    mathwonk

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    well i just looked up stirling and it seems to suggest at a quick calculation, not guaranteed, that this limit is e/sqrt(2pi)
     
  5. Aug 2, 2004 #4
    e/(2pi)^(1/2) aprox= 1.0844

    my calc can do the limit up to 200 and it equals about 2.67021... that's why i thought it may = e
     
  6. Aug 3, 2004 #5

    Galileo

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    With Stirlings approximation: [itex]N!\approx N^Ne^{-N}[/itex], you indeed get:

    [tex]\frac{N}{(N^Ne^{-N})^{\frac{1}{N}}}=\frac{N}{Ne^{-1}}=e[/tex]
     
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