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

Question

The YouTube video "Infinity Factorial" by BiBenBap says that $\infty! = \sqrt{2\pi}$.

First of all, isn't the factorial operator an arithmetic operator? It is just multiplication following more specific rules. So, if the factorial operator is an arithmetic operator, then $\infty! =$ undefined, since arithmetic on infinity isn't allowed.

Past that, if we write out the factorial, then the answer is still undefined:

$$\infty \times (\infty -1) \cdots (\infty - (\infty -2)) \times (\infty - (\infty-1))$$

Given that any number substracted/added to infinity is infinity, then the last factors of the factorial are zero. $\infty \times 0 = $ undefined. If we move the parantheses however, those subtractions become additions, meaning we've got $0$, plus some number. Well, if that's allowed (which it isn't, since arithmetic isn't allowed on infinity), then we just get infinity times and infinite number of infinities, times an infinite number of finite numbers. That would obviously be infinity.

The math used in the linked video is far beyond me. Could someone explain how that video is wrong/right and how I am wrong/right?

Answer 1

The final result in the video is not wrong. But what much of the video says/does is wrong. $\infty !$ is a nonsensical symbol. Think of this as any other advertisement you see on TV: it may have some real things, but a lot of it is phrased misleadingly/deceptively to get people to click/view it.

The factorial is an arithmetic operation defined only on non-negative integers (as you have noted in your second sentence), and this definition is done recursively, i.e we define

• $0!= 1$ by definition
• For any $n\geq 0$, we define $(n+1)!= (n+1)\cdot n!$

So, we can only take factorials of non-negative integers $0,1,2,3,4,5,6,\dots$

Trying to somehow claim that $\infty !=\sqrt{2\pi}$ is then a nonsensical assertion based on the above definition (because $\infty !$ is not even defined based on our above definition).

---

The correct statement.

The correct statement (i.e no wishy-washy handwaving nonsense) is that Riemann's zeta function is such that $\zeta'(0)=-\frac{1}{2}\ln(2\pi)$. What does this mean? You say that the math used in the video is far beyond you, so fine, let me just tell you parts of "what" is being said.

Riemann was a famous mathematician, and one of his works was about a certain function. This function has the symbol $\zeta: \Bbb{C}\setminus \{1\}\to\Bbb{C}$, and it is what we now call "Riemann's Zeta function". What does this function do? Well, it takes a complex number $s\neq 1$ as an input, and it outputs a certain complex number $\zeta(s)$. The exact definition of $\zeta$ is not really important to us right now (and indeed giving a correct definition for this is something one can only do after a half-semester course in complex analysis). All you need to know is that one can prove that the function $\zeta$ is differentiable at $s=0$, and that \begin{align} \zeta'(0)=-\frac{1}{2}\ln(2\pi) \tag{$*$}. \end{align}

So, the correct statement is $(*)$. This is 100% correct mathematics (I didn't fully watch the video, so I won't comment on whether the Youtuber's explanation of this result is correct). But, rest assured that $(*)$ can be given a completely airtight proof. HOWEVER, what is utterly nonsense is going from $(*)$ to the statement that $\infty!=\sqrt{2\pi}$. This is what I was talking about in my first paragraph: there is some real mathematical content in this video (i.e the definition of $\zeta$ and the calculation of its derivative at $s=0$). However, the video is also full of "false advertisements" in that it makes absurd claims like $\infty!=\sqrt{2\pi}$.

One has to really turn a blind eye to several mathematically incorrect steps (such as calling $P=1\cdot 2\cdot 3\cdots$, which is really just $\infty$, and doing arithmetic with that), and then hand wave alot of details to even heuristically try to justify the relationship between the correct equation $(*)$ and the nonsensical $\infty!=\sqrt{2\pi}$.

---

Final Remarks.

The zeta function is also notorious in "everyday math" claims, because of the somewhat (in)famous $1+2+3+4+\cdots =-\frac{1}{12}$ nonsense. Anyway, to finish off, the statement that $\infty!=\sqrt{2\pi}$ makes as much sense as saying "the earth and moon are both round, therefore, the earth and moon are the same thing".

Answer 2

Most of the video is irrelevant, as it’s only about calculating the value of $\zeta'(0)$. The shady part is actually happening in 4:33, as he is ignoring the rules when differentiation may be pulled into a series. As $\lim\log(n)=\infty$ it is impossible for that series not to be $\infty$.

Basically we have a sequence of functions $f_k$ and take the series $\sum_k f_k$. Supposing that this converges we then want to take the derivative $\frac{\mathrm d}{\mathrm d x}\sum_k f_k(k)$. What this guy then is doing is to say $$\frac{\mathrm d}{\mathrm d x}\sum_k f_k(x) = \sum_k \frac{\mathrm d}{\mathrm d x}f_k(x) $$

Indeed we have a Theorem that states that this is true for all points in an interval $[a,b]$ under the conditions that the series of the derivatives converges uniformly.

But this is not true for the functions he is using!

So this is a case of: If we implicitly assume that the surprising property $P$ is true we can suddenly prove that the slightly weaker property $Q$ is true.

Answer 3

No. The video is wrong. The regularized value of this infinite product is different. See the details here.