I Found Out What Infinity Factorial Is

What is Infinity Factorial equal to? You might be thinking there is no infinity factorial value. In some sense, you intuition is true. BUT what happens when we break out the Riemann zeta function and the Dirichlet eta function?!

Let's see why infinity factorial = sqrt(2pi)

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Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information.

#math #brithemathguy #infinityfactorial

[BriTheMathGuy]

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[knutthompson]

The analytic continuation can be useful for many purposes, but any time you are leveraging the Riemann Zeta function outside its domain, it is no longer the same as Zeta function. So the results, at best, need some huge asterisks.

[EduardoBatCountry]

Aaaah yes… the same analytic continuation which tells that: 1+2+3+…=-1/12
I’m not gonna fall in this trick again

[luizestilo]

For all of the kids watching, know that
he is practicing the dark arts, forbidden spells that have been cast away dozens of years before us.
The type of magic prohibited by the whole council to ever be put into practice, that you'll only hear from shady men roaming in dark corners of the town.






Also, don't show this to your math teacher, they may have a stroke.

[alphalunamare]

It is certainly entertaining but lacks rigour .. I say that as a 'cop out' because I really enjoyed the presentation, just don't believe it :-) I am wondering however about how this sort of thinking might shed intuitive light on The Prime Number Theorem?

[douglasstrother]

I was really looking forward to "42".

[hjdbr]

Really interesting video! As other people commented, I should point out, due to the flashbacks from numberphile's -1/12 video, that all of this manipulation is more a sleght of hand than a "mathematical proof that infinity! = sqrt(2pi)". infinity! = infinity and that's it. This is basically some manipulation to try to assign an "alternative value" to something that should have been infinity, but it's not its actual value. With that being said, I really enjoyed the interesting manipulations done in the video, and was absolutely not expecting the zeta function or pi to show up!

[SuperMerlin]

I thought this was going to use the fact that n! is equal to the size of the set of all permutations of a n item set. infinity factio is the size of the set of real numbers.

[РођакНенад]

I'm surprised by the number of infinite products or sums whose solutions involve pi. it seems to be involved with everything in math, which I find very intriguing.

[dlevi]

Yes, the result in terms of association is there, as are the deep connections between the factorial as a "probabilistic" function and the Gaussian integral - which lo and behold has an area of √π between -∞ and +∞

However, switching the order of differentiation and summation and moving/grouping elements on non-absolutely convergent series, using analytic continuation in place of the original function and so on are all sleight of hand that is IMHO very dangerous to present as 'always motivated and possible'. It's a bit like the 'proof' that the sum of all natural numbers is equal to -1/12 just by using (unjustified) series manipulation.

I understand you set yourself a 5 minutes target, but this is not one of your best videos. At the very least it should be very heavily caveated.

[Rudxain]

sqrt(tau) also shows up in Stirling's Approximation of the Gamma Function. It still appears in the improved version (Gosper's Approximation)

[dirichlettt]

Every second of this video hurt my very soul, but it is a great presentation nonetheless!

[kasuha]

Some time ago I followed the "proof" that sum of all natural numbers is equal to -1/12 and my observation was that the proof was rather that sum of all natural numbers plus infinity equals to -1/12 plus infinity. And after we subtract infinity from both sides, we get the "result". I don't expect this to be any different but I'm lazy to go through that process again.

[aaronhendrickson]

Note Stirling's approximation log(n!)~n log(n)+log(n)/2+log(sqrt(2 pi))+O(1/n). If we ditch the diverging terms as n->inf then we get the same result as in the video.

[romajimamulo]

Yeah this ... Has a few problems. We got the zeta function by working with the definition, but then we used the continuation instead. The actual zeta function, the original one, diverges at the point we're looking at

[douglasstrother]

I need to incorporate this when computing project costs.

[digxx]

It's nice and entertaining, but you definitely can not just do that addition and subtraction of a divergent quantity at 3:15.^^

[frederikl.jatzkowski]

Around 3:15 you add two diverging series. This can only be done with converging series without altering the value of their sum.

[amangandhi]

@BriTheMathGuy
at 1:00, n^0 is equal to 1 is not true when n equals infinity, yet the substitution is made inside the summation where n takes values from 1 to infinity.

this mistake is connected to this inconsistent "woo woo" result of infinity factorial is square-root of 2 pi.

[parthhooda]

2:55 This is a problem. By analytic continuation he means extending the function beyond it's domain. It does make the function nicer but still the original definition of function which was sum of reciprocals of powers of natural numbers makes it clear that reiman zeta function for 0 would actually diverge as it would mean adding 1 infinite times. So riemann zeta function at 0 is -1/2 but the sum that you proposed and defined as riemann zeta function does not equal -1/2 for 0.