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Factorial of infinity

  1. Feb 13, 2013 #1

    ShayanJ

    User Avatar
    Gold Member

    I was studying about infinite products that I got to the relation below in
    http://mathworld.wolfram.com/InfiniteProduct.html

    [itex]
    \infty != \sqrt{2 \pi}
    [/itex]

    It really surprised me so I tried to find a proof but couldn't.
    I tried to take the limit of n! but it was infinity.Also the limit of stirling's approximation was infinity.
    So what?Is it correct?if yes,where can I find a proof?
    Thanks
     
  2. jcsd
  3. Feb 13, 2013 #2

    lurflurf

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    Homework Helper

    That is not for the usual product, but for regularized products.

    in general (I use a ^ to denote regularized products as is sometimes done)
    $$\prod_{n=1}^{_\wedge ^\infty} \lambda_n=\exp (-\zeta_\lambda ^\prime (0)) $$
    where
    $$\zeta_\lambda (s)=\sum_{n=1}^\infty \lambda_n^{-s}$$
    then for you example lambda_n=n
    $$\infty!=\prod_{n=1}^{_\wedge ^\infty} n =\exp (-\zeta ^\prime (0))=\sqrt{2 \pi}$$
     
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