# Factorial of infinity

by Shyan
Tags: factorial, infinity
 P: 490 I was studying about infinite products that I got to the relation below in http://mathworld.wolfram.com/InfiniteProduct.html $\infty != \sqrt{2 \pi}$ 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
 HW Helper P: 2,078 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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