Under what conditions, or how can be done that, one says that something divergent can have a value?

Question

Excuse my question, I just don't get it.

In this proof, is mentioned:

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!)$ .

That is, one ultimately gets $ - \lim_{n \to \infty} \ln(n!)$

In that same proof, mentions that $\zeta'(0)=-\ln\sqrt{2\pi}$

Ultimately, one would get to "$\infty!=\sqrt{2\pi}$"

Why the two values "$\infty$" and $\sqrt{2\pi}$ were assigned to be equal?

Thanks in advance.

Answer 1

Two concepts need to be understood:

• analytic continuation
• formality

In mathematics when something is done "formally" that means it is not to be taken literally and no claim of logical rigor is made.

Thus $\zeta(s) = \sum_{n=1}^\infty 1/n^s $ is true if $s>1$ when convergence of series is defined the way you saw it defined in secondary school. But this function can be analytically continued so that there is a holomorphic function $\zeta$ defined everywhere in $\mathbb C$ except at some isolated points, and one point in that domain is $-1,$ and $\zeta(-1) = -1/12,$ so then one formally writes $\sum_{n=1}^\infty 1/n^{-1} = 1+2+3+\cdots = 1/12.$ But that doesn't mean that series converges to $1/12$ in the usual sense, nor necessarily even in any of the someone less usual senses.