How is it possible that $\infty!=\sqrt{2\pi}$?

I read from here that:

$$\infty!=\sqrt{2\pi}$$

How is this possible ?

$$\infty!=1\times2\times3\times4\times5\times\ldots$$

But \begin{align} 1&=1\\ 1\times2&=2\\ 1\times2\times3&=6\\ &~\vdots\\ 1\times2\times3\times\ldots\times50&=3.0414093201713376\times10^{64} \end{align}

This is obviously increasing, so how does factorial of $\infty$ become $\sqrt{2\pi}$ ?

Solution 1:

It is taken from $$ 1\cdot2\cdot3\cdot \ldots \cdot n= n!$$ This is the exponential of $$ \ln(1)+\ln(2)+\ln(3)+ \ldots + \ln(n) = \ln(n!) $$ Now if you write formally the derivative of the Dirichlet-series for zeta then you have $$ \zeta'(s) = {\ln(1) \over 1^s}+{\ln(1/2) \over 2^s} +{\ln(1/3) \over 3^s} + \ldots $$ This is for some s convergent and from there can be analytically continued to $s=0$ as well from where the the formal expression reduces to $$ \zeta'(0) = -(\ln(1) +\ln(2) +\ln(3) + \ldots )$$ which is then formally identical to $ - \lim_{n \to \infty} \ln(n!)$ .

Now the slope of zeta at zero can numerically be approximated and gives a concrete number $\zeta'(0) \approx -0.91893...$. It can also analytically be determined to equal $\ln(1/\sqrt{2\pi})$ .

Finally, since the formal notations coincide (except of the sign) one goes to write the exponential of this value to become the "regularized" value of the infinite factorial.