The video explains how an infinite product, seemingly divergent, surprisingly converges to a finite value. Initially, the product $1 \times 2 \times 3 \times \dots$ to infinity is presented as diverging. However, the video demonstrates a method to find its actual value, which is $\sqrt{2\pi}$. The technique involves taking the natural logarithm (ln) of the product. The natural logarithm of a product is the sum of the natural logarithms of its terms: $ln(1 \times 2 \times 3 \times \dots) = ln(1) + ln(2) + ln(3) + \dots$. The core of the problem then becomes evaluating this infinite sum. The video introduces the Riemann zeta function and its derivative at $s=0$ as the key to solving this sum. Specifically, the negative of the derivative of the Riemann zeta function at $s=0$ is equal to the sum $ln(1) + ln(2) + ln(3) + \dots$. The value of this derivative is given as $-\ln(\sqrt{2\pi})$. By combining these steps, the original infinite product is shown to be equal to $e^{ln(\sqrt{2\pi})}$, which simplifies to $\sqrt{2\pi}$. This result is presented as a counter-intuitive and "crazy" outcome.

Mathacy

So, 1 times 2 times 3 times dot dot dot all the way to infinity surely diverges to infinity. But actually, this has an answer of square root 2 pi.

And you may drop down and be like, "Well, how?" And what we do is we take the original product and then take e to the ln of that product, which is equal to e to the ln one plus ln two plus dot dot dot.

Now, essentially, what we need to know is what is ln one plus ln two plus dot dot dot. Now, we can solve this using the super famous Riemann zeta function, specifically its derivative at s equals zero, which is the negative of the sum that we want.

And since the derivative is the gradient, then calculating the gradient at s equals zero gives minus ln square root 2 pi.

Now, combining everything together gives 1 times 2 times 3 times dot dot dot equal to e to the ln square root 2 pi. And since e to the ln square root 2 pi equals square root 2 pi, then we have our crazy results.

Math is BROKEN: 1x2x3x...= √2π

so 1 times 2 times 3 times dot dot dot
all the way to infinity
surely diverges to infinity but actually
this has an answer of square root 2 pi
and you may drop down and be like well
how and what we do
is we take the original product and then
take e to the ln
of that product which is equal to e to
the ln one plus l and two plus dot dot
dot now essentially what we need to know
is what is ln one plus ln two plus dot
dot dot now we can solve this using the
super famous
riemann zeta function specifically its
derivative
at s equals zero which is the negative
of the sum that we want and since the
derivative
is the gradient then calculating the
gradient at s equals zero
gives minus ln square root 2 pi now
combining everything together gives 1
times 2 times 3 times dot dot dot equal
to e to the ln square root 2 pi and
since e to the l and square root 2 pi
equals square root 2 pi then we have our
crazy
results

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Math is BROKEN: 1x2x3x...= √2π

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