Register to reply 
Factorial of infinity 
Share this thread: 
#1
Feb1313, 10:44 PM

P: 923

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
Feb1313, 11:41 PM

HW Helper
P: 2,264

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 $$\infty!=\prod_{n=1}^{_\wedge ^\infty} n =\exp (\zeta ^\prime (0))=\sqrt{2 \pi}$$ 


Register to reply 
Related Discussions  
Integrating sinc(x)^4 between negative infinity to infinity using complex analysis  Calculus & Beyond Homework  6  
L'Hôpital's rule for infinity minus infinity  Calculus  0  
A Definite integral where solution. involves infinity  infinity  Calculus & Beyond Homework  8  
A Definite integral where solution. involves infinity  infinity  Calculus & Beyond Homework  3  
Lim n to infinity for factorial  Precalculus Mathematics Homework  2 