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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/ @brithemathguy
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]

🎓Become a Math Master With My Intro To Proofs Course! (FREE ON KZbin) kzbin.info/www/bejne/aZTdmJl-irGNedU

[@Fire_Axus]

No

[@EduardoBatCountry]

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

[@simongross3122]

But apparently that spurious result is quite important to string theory.

[@dlevi67]

@@simongross3122 Not just to string theory. And the result is not spurious; there are precise ways to justify it; none is (truly) elementary, unfortunately.

[@simongross3122]

@@dlevi67 It is rubbish. There are ways to make any non-convergent series apparently equal anything. Still, if you want to believe it, I can't stop you.

[@dlevi67]

@@simongross3122 It is not rubbish and it is - or can be made - rigorous. Not in the way (e.g.) the Numberphile video from a few years ago did, but Mathologer did a much better work (for example). Still, if you don't want to believe it can be made rigorous, I can't stop you. 😁 There is a theorem (Riemann's rearrangement theorem) that proves that a certain type of series can be made to converge to any value, but not "just any non-convergent series".

[@simongross3122]

@@dlevi67 I wonder if Bri the Math Guy can settle this.

[@knutthompson7879]

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.

[@BriTheMathGuy]

Good point!

[@kazedcat]

Except when they use it for calculation in quantum physics then it becomes valid application.

[@sontapaa11jokulainen94]

@@kazedcat bruh 😂

[@opalb9006]

uhh math math, um zeta alpha pi tau stuff i understand 100% definitely uh i dont feel dumb at all right now

[@xGOKOPx]

@@opalb9006 It would cost you exactly $0 not to write that comment

[@luiz00estilo]

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.

[@BriTheMathGuy]

Did you ever hear the tragedy of Darth Plagueis the Bris?

[@好吧-h6k]

I understood everything until whatever that dirichlet eta thing is

[@Chadniger]

Damn y'all seriously assume all kids have the same mindset

[@watermocules7735]

@@Chadniger ever heard of a joke

[@phenixorbitall3917]

Haha ^^

[@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?

[@BriTheMathGuy]

Oh it definitely lacks rigor :) Glad you enjoyed it anyway!

[@georgestrvanger6878]

This uses the Reimann Zeta Function. Every result is gonna lack rigor. It's like saying 1+2+3+... = -1/12

[@alphalunamare]

@@georgestrvanger6878 :-)

[@Jehannum2000]

Maths is enjoyable when you leave out the rigour.

[@mjmulenga3]

@@Jehannum2000 as are most pursuits.

[@douglasstrother6584]

I was really looking forward to "42".

[@BriTheMathGuy]

😅

[@hqTheToaster]

Don't tell them about 2^2^x^2^x^1/pi as x approaches infinity.

[@SuperMerlin100]

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.

[@isospectral3537]

Aleph null factorial, more specifically.

[@farrankhawaja9856]

@@isospectral3537 Lol aleph null factorial is actually a mindblowing concept because it is a weakly inaccessible cardinal so you can't really subtract from it and ever get a finite result. Nice.

[@angelmendez-rivera351]

@@farrankhawaja9856 You cannot subtract cardinals, period, so that is redundant.

[@hjdbr1094]

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!

[@angelmendez-rivera351]

*Infinity! = Infinity and that's it.* Really, now? Can you prove that this is true? You cannot. You would say that lim n! (n -> ♾) = ♾, but you are assuming lim n! (n -> ♾) and ♾! represent the same thing. You cannot prove that, you are just making an assumption. This is just as much a sleight of hand as the video itself. *This is basically some manipulation to try to assign an "alternative value" to something that should have been infinity, but it's not actual its actual value.* How do you know it is not its actual value? What determines the actual value of ♾!, then? I wish I had the self-control to refrain from ranting about how much I hate it when people pretend to be well-informed about a topic, only to very thoroughly demonstrate that they are not. Your comment reveals that what little knowledge you have on the subject comes entirely from reading people's comments about Numberphile's video, instead of having read any actual mathematical sources on the matter. And the dismissal is, as some would put it, "all talk, no bite." You have not appealed to a single mathematical concept or any formal theory or provided a single meaningful correction. Your deliberation boils down to "this is wrong because it is wrong." Meanwhile, there actually is a lot of work that shows that these results are indeed perfectly justified. These results are used in physics. But even beyond just talking about physics, it has plenty of meaning in various disciplines in pure mathematics. I would recommend that you do some reading on the theory of Banach limits.

[@nerdsgalore5223]

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

[@SjS_blue]

I feel like this video was made to get people interested in math and the wierd things that can happen sometimes. yeah. for example every time there is an infinity symbol instead of a limit. so many hand-waving arguments, it hurt my brain

[@dlevi67]

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.

[@BriTheMathGuy]

I really appreciate your honest feedback. I'm always trying to improve my videos so I hope you will continue to provide constructive criticisms in the future. Thank you for watching!

[@dlevi67]

@@BriTheMathGuy Thank you - and best wishes for the Holidays!

[@BriTheMathGuy]

@@dlevi67 You as well! :)

[@kazedcat]

Analytic continuation is enough of a disclaimer.

[@Rudxain]

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

[@BriTheMathGuy]

Cool!

[@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.

[@oyungogdfrust4136]

2pi is tau, its sqrt(tau)

[@danielarnold9042]

@@oyungogdfrust4136 what's tau used for?

[@oyungogdfrust4136]

@@danielarnold9042 almost every equation you use 2pi in is a lot simpler with tau used instead of 2pi, for example the classic 2piR is just tauR with tau

[@danielarnold9042]

@@oyungogdfrust4136 ok thanks

[@malikyamin7747]

π^2 = 9.8(g)

[@douglasstrother6584]

I need to incorporate this when computing project costs.

[@parthhooda3713]

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.

[@digxx]

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

[@ChannelOfElveman]

Yeah, since it's a weak convergence, you can basically get a number as big or small as you like as a result of those manipulations

[@amangandhi2863]

@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.

[@amangandhi2863]

one way to analyse such inconsistencies is to locate the exact step where the value changes, in this case what is the transition step where the expression considered changes value from infinity to sqrt(2pi) . the first few steps have value infinity, the last few steps have value sqrt(2pi), so which step makes the transition possible? analysing possible mistakes at that step.

[@Lucaazade]

x^0 is definitely equal to 1 for any object represented by the symbol x :)

[@amangandhi2863]

@@Lucaazade except infinity, infinity raised to zero is 'not-defined' and definitely it is not 1

[@amangandhi2863]

also zero raised to zero is not 1

[@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.

[@Qermaq]

Well, ya. It's no different than saying 2 = 3 since 2*0 = 3*0. I'm fond of creativity in math, but not when it's presented as having any tendrils to reality. Simply put, if the answer is inconsistent with the prediction, you have made a mistake in the answer or in the prediction.

[@NikeDattani]

@kasuha , @Qermaq 1+2+3+4+... Is a *divergent* series. However there's ways of assigning numbers to divergent series, just like the Cauchy Principle Value is a way to assign a value to a divergent integral. If you assign the divergent sum a value based on the Ramanujan summation convention, you'll get -1/12. I'm considering to make a video about it next week if there's enough interest. I'm a fairly new KZbinr!

[@Qermaq]

@@NikeDattani I assigned names to the house plants. The ficus was assigned the name Larry. The fern was assigned Janet. But is there any VALUE to us in naming the house plants? Or tacking a number onto a divergent series?

[@NikeDattani]

@@Qermaq that's a very good point. However Ramanujan's sum is not completely arbitrary, there's a few different justifications for it. You don't get -1/13 with that sum and the Riemann zeta function, you get -1/12. The y-intercept for a certain plot related to that sum, has slope = -1/12, not infinity or +1/13 or something else. Furthermore it has its "value" when studying the Casimir effect!

[@Qermaq]

@@NikeDattani I vaguely recall bprp did a few videos where he shows 1 + 2 + 3 + ... can take other values like -1/8 and -1/9. Can't find them at the moment. So your video would have to convince me not that -1/12 *can* be a value of the sum, but *should* be the value.

[@frederikl.jatzkowski7743]

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

[@TheTruemanBoi]

That's the idea behind these operations, of course these series are divergent, but just as an experiment, the idea is treat them as convergent series, it makes no sense, and of course the result obtained is not the true result of these operations, that's why you can obtain -1/12 of the addition of all natural numbers. Addition which diverges and of course has no sense as there is no decimal neither negative number to obtain such result.

[@farfa2937]

I think this is more of a "you converge because I said so" scenario.

[@raphaelnej8387]

this is basically trying to compute S by making S = S + S - S and say yo look at this S + S it s actually the log of the wallis product which is equal to ln(pi/2) so S = S + S - S = ln(pi/2) - S so 2S = ln(pi/2) so S = ln(pi/2)/2 the prank is that S was a serie adding and subtracting bigger and bigger quantities which is mathematically illegal

[@angelmendez-rivera351]

@@raphaelnej8387 It isn't mathematically illegal at all. Banach limits.

[@angelmendez-rivera351]

@@TheTruemanBoi Why should it matter that there is no decimal or negative quantity? This is a series, not an addition of finitely many elements. Did you know that you can have a series of infinitely many rational numbers that converge to π? According to your incorrect assumptions, that should be impossible.

[@Senshidayo]

That was interesting but all of these videos that use analytic continuations don’t list any caveats that occur when you extend the generalizability of a function. Will the new function work in all cases that it did before? Not necessarily. Especially since you start off with a factorial… it was fun but getting sqrt(2pi) really is a hint that you are looking at a completely different object than what the original function was built for, given the argument.

[@dlevi67]

Not only that, but the most commonly accepted extension of the factorial is the Euler Gamma function, which lo! diverges at infinity...

[@biblebot3947]

@@dlevi67 the gamma function isn’t the analytic continuation of the factorial. It’s just one chosen generalization. Analytic continuations are unique for each function and the factorial isn’t continuous, let alone differentiable, let alone smooth, let alone analytic.

[@dlevi67]

@@biblebot3947 Sorry - should have said extension - corrected; the gamma function can be analytically extended to the whole complex plane, but that is irrelevant here. (Love the autocorrect substituting a Belgian town for a Greek letter!)

[@MultiRobotnik]

1:06 the _negative first_ derivative, or the negative _first derivative?_ Wouldn't the "negative first" derivative be the integral?

[@hamzasalikmaths9794]

There are a mistake. When you calculated the derivative of the eta function you had put in the symbol sigma (-1)^n then normally it is (-1)^(n-1)

[@RafaxDRufus]

This is really interesting, loved the video. However, it may be a little misleading for all those people who don't have sufficient knowledge to recognize the weird things that happen when you work with analytic continuations and rearrangement of non convergent infinite series. Even though you make the right comments in the video, I think it would be better to put emphasis on it, or leave a small text in the video reminding it. Otherwise you can contribute to more people thinking that sum(n) = -1/12

[@Astrodude4494]

I'm in 8 class n I'm still trying to understand these things 🤣

[@nguyenthai3140]

n! can be written as: 1*10*100*...*10^a*something, but the digit of 1*10*...10^a turn to ...00000 when a goes to infinity, which is (p-adicly) equal to 0. So (infinity) ! in some way, is equal to 0

[@nylonco7134]

While I wouldn't go shouting from the rooftops that infinity! = sqrt(2π) (lack of rigour will do that), this kind of dangerously informal math seems like it serves a great purpose in demonstrating relationships that *might* have something deeper going on, with the consequence of motivating a more rigorous follow up study. So, still some potential as a useful result (albeit indirectly), but dang did the lack of formality knock me to the ground. If you were going to make this video, I'm not sure there are many better ways to do it. Hats off, partly on account of me having been knocked to the ground, as mentioned previously. If I'm going to criticize anything, I can say I would have appreciated a final word at the end addressing the absolutely horrifying lack of formality, but that's about it.

[@m3morizes]

I don't understand why everyone in the comments is so angry. The whole point of this video is to use (strictly incorrect) reasoning to playfully assign a value to an obviously undefined, intriguing expression.

[@severnkariuki9129]

What's really entertaining is that infinity factorial and √2π are all important concepts to link.

[@DubioserKerl]

So, is this some kind of (sum of n = 1 to ininity = -1/12) stuff or what?

[@BriTheMathGuy]

It certainly has that kind of feel to it!

[@bleep0004]

@@BriTheMathGuy that most probably means you broke a lot of math rules to get to this solution. I'm not a math expert so I can't tell you where you were wrong.

[@bleep0004]

@@BriTheMathGuy kzbin.info/www/bejne/a4SpeZuqpKmLhbs This shows up as the first video about this. You're the 5th misleading KZbinr using the same calculation.

[@SotirisSimos]

It's exactly like that, maybe worse

[@aaronhendrickson3013]

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.

[@davidchung1697]

A good observation!

[@shruggzdastr8-facedclown]

@Bri: Is there any way that Ramanujan's -1/12 (his result for the infinite sum of all pos. integers) has a relationship with this infinite factorial result? In other words, can this be "solved" using -1/12?

[@ronaldtownsend5745]

Yes, there is a definite relationship between the two results. Both use Ramanujan's Theorem outside its specified domain.

[@manucitomx]

Thank for a great video. Keep up enlightening us. Happy holidays to you.

[@BriTheMathGuy]

Same to you!

[@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

[@johannesvanderhorst9778]

Because it diverges, one considers the analytic continuation, where on 0 is regular. So when taking derivatives, the analytic continuation also is regular at 0.

[@Rolancito]

1:12 thank you for apologising in advance for breaking math, I was already getting offended at your result

[@aokkishi5064]

Everything is good but even if you invoke the Analytic Continuity argument, zeta prime of zero is not that sum, it has another expression, that expression for the zeta function is only valid if s>1.

[@aokkishi5064]

But if you want some cool results that are not rigorous, in the way you just calculated zeta(0)=-1/2. Zeta(0) based on what you doing Mr. Bri is the sum 1+1+1+1.. So yeah you just proved that 1+1+1+1..=-1/2

[@NerdWithLaptop]

1+1+1…=1+(1+1)+(1+1+1)… = 1+2+3…=-1/12 -1/12 = -1/2 12=2 6=1 10=5 1=2

[@BriTheMathGuy]

It's a cool sum that deserves its own video!

[@aldobernaltvbernal8745]

@@NerdWithLaptop **casually rearranges divergent sum**

[@shehnazsalahuddin6053]

Wow this is so interesting! I love this video!

[@BriTheMathGuy]

Glad you enjoyed it!

[@erickehr4475]

I have a proof that infinity factorial is 0: Clearly it is an integer as it is a product of integers. Now, given any natural number n, the nth digit (counting from the right) of infinity factorial is 0 as 2^n.5^n is a factor. So since every digit is 0, the number itself is 0. QED

[@nikolson191]

Integers are finite by definition, so the product of an infinite number of integers > 1 is not an integer.

[@WestExplainsBest]

Square root of 2 pi would have never been my guess...mind blown!

[@ben_adel3437]

Can infinity factorial even exist when you put it in the gamma function it just goes to infinity and it does make sense considering the Γ(x)=xΓ(x-1) rule but it doesn't feel right

[@MichaelGrantPhD]

You certainly do a *little* more qualification here than Numberphile's infamous harmonic series video. But in my view it is nowhere near enough to justify the development-certainly not enough to justify the bare equality in the final frame of your video or its unqualified declaration in your conclusion. To further reinforce my point, consider the summation form that you equate to the Riemann zeta function. This infinite sum is divergent at s = 0; it does not have a finite value. You cannot say it equals the Riemann zeta function at s = 0; it does not. The two quantities coincide only in the region of convergence of the sum. Therefore, you cannot use that sum to define the derivative of the Zeta function outside of that region.

[@ProCoderIO]

That paper by Terry Tao really sheds light on these non standard values associated with unconventional sums and what not.

[@andrewkarsten5268]

Bruh everyone quit getting so caught up in the lack of rigor. I get it, I really do, I’m majoring in math at uni right now, but this kind of fun little stuff is my favorite part, because you learn stuff when you push math right to it’s limits and beyond, not when you stay well behaved! If you want to be right “in the usual sense”, the infinity!=infinity, plain and simple. Yes we’re using analytic continuation (and a few other things but the continuation is kind of the most pressing here), but don’t forget that continuation gave us 1+2+3+4+...=-1/12. Again, if you want to be “usual”, then that sum blows up to infinity, simple, end of discussion, but then again using this result has been incredibly useful in many other areas of math and programming. Idk, just maybe give it a chance. Remember that when i was created, everyone thought it to be lunacy because “you can’t have that”. How are we to say this won’t lead to the next big discovery? Just chill. Yeah we all know it lacks rigor, we don’t need hundreds of comments all saying how terrible it is because of that

[@law26504]

Well the gamma function would certainly disagree very much with this answer.

[@BriTheMathGuy]

🤔

[@VPWedding]

So if you have an infinite number of objects, there are 2.5 ways to arrange them? Can you list them for us?

[@saravanarajeswaran2626]

0:23 " Assuming that we can use properties of logarithm" yep that's where it went wrong

[@cythism8106]

Imagine you had infinite spaces and infinite blocks that are numbered sequentially that go in those spaces. How many unique ways can you arrange those spaces? According to this, it's less than 3. Logic 1000000

[@RB-ew6lo]

I laughed so hard at this, spot on summary :-)

[@prépa_X]

I read from Wikipedia that the analytic expression of the Zeta function that you used, concerns complex numbers s whose Re(s)>1; So we can't use this formula with s=0

[@erictart4225]

This feels like a good example for Gödel's Incompleteness Theorem. Stringing together disparate mathematical topics seems almost like programming in a sense where the underlying code dependencies might conflict. Seeing people mention domains of applicability reminds me of the "warnings" that code compilers spit out at us. And which we promptly ignore.

[@sowndolphin5386]

finally something that makes sense after 1+2+3+4..... = -1/12 thing

[@lemniskate_ayd]

With the Zeta function, you can also prove the very famous Σ of natural numbers = -1/12 … so… I’m not that surprised😂

[@longlostwraith5106]

The sum equals infinity. The Zeta function value of the sum equals -1/12.

[@magicmulder]

But still, at the end of the day this is like all those ways in which you can show that "infinity divided by infinity = any number you want".

[@imsounak19]

We can use the Gamma Function as well

[@core3gamegd587]

there are other answers. such as there are OTHER infinities. if we assume where dealing with ℵ_0 (aleph null or the amount of natral numbers on a number line) then we can get another answer. if the continum hypothysis is true, aka 2^ℵ_x=ℵ_x+1 we can aproxomate it to be >ℵ_1, but is actually ℵ0^ℵ0. so maby the root(2pi) is for absolute infinity? but idk, maby we are both right, or both wrong, or something else.

[@PPppp-te5qs]

Negitive derivative? That is impossible to have: You can not count negative. 1:24 Negativ derivata? Det är omöjligt att ha: Du kan inte räkna negativt. 1:24

[@thatdude_93]

I can't even see the video through all this handwaving

[@BriTheMathGuy]

I think the video is in there somewhere 🤔

[@rythmx123]

if you take a closer look π is somewhat infinity itself.. you can use 3 radius of a circle to cover its circumference, but still the remaining space is infinite, that is π

[@rururu5877]

We have perfectly defined pi, as in some magic paint shop you can ask for pi units of paint. The problem with pi is the same as five years old knowing some Mister. Pi but for rhe love of god aren't able to draw him well But it's true that pi has some cool things going on

[@bernardfritts4173]

@23 seconds.... Holy cow im excited.

[@tykjpelk]

This almost looks like a proof by contradiction against analytic continuation.

[@bobnewman6196]

This seems interesting and I think there is no shame in extending this topic to a much longer video. I wish the explanation of the proof was a little more thorough. It seems like 15% of the video was advertisement. However I did find the topic very interesting!

[@PrimusProductions]

Any relation to the area under the normal distribution being sqrt(pi)?

[@orphixigl1476]

If we consider 1,2,3,... as cardinals, then the product 1×2×3×... is 2^aleph_zero.

[@eye2077]

and aleph zero is pretty much infinity I think.

[@davidblauyoutube]

Did you know that all of the (complex) roots of the (analytic continuation of) the zeta function have real part 1/2? I have a marvelous proof of this, but unfortunately this comment section is too small to contain it.

[@kjl3080]

Managed to reference both FLM and Riemann hypothesis nice

[@johannesvanderhorst9778]

I believe that the proof, if it exists, will be too long for the comment section. I don't believe yet that you have a proof, as I know no better than it is still a hypothesis.

[@PPppp-te5qs]

Comment l’infini est-il factorisé en nombre positif ?

[@PPppp-te5qs]

How is infinity factorial a positive number?

[@PPppp-te5qs]

Die Unendlichkeit ist unvorstellbar groß. Warum ist diese Zahl dann real, wenn die Unendlichkeit keine Zahl ist? 0:25 Infinity is unimaginably large! When you factorial it, it is big Why? 0:25

[@Thitadhammo]

Well.. analytic continuation is all well and good for some purposes but what you are really doing is some handwavery that results in assigning a value to a series that might or might not have some meaning. It's evidently not true that multiplying all natural numbers together IS sqr(2pi). Others have already commented on the lack of rigour. Your presentation is fun to watch, though. It really helped me stay in bed for just four more minutes.

[@BriTheMathGuy]

Oh it certainly lacks rigor - but it's fun :) Thanks for watching and commenting!

[@Thitadhammo]

@@BriTheMathGuy Oh, for sure it is!

[@christressler3857]

It's almost like a shadow from unforeseen math lurking.. *around* the Riemann sphere (having to multiply around and through infinity brings you back to the affine)

[@tobiasreckinger2212]

The moment I hear someone mentioning the Riemann Zeta function and analytic continuation in the same sentence I know that the result won't make any sense from a practical point of view

[@caclesi]

Yeah I thought the same

[@kazedcat]

Every time you deal with infinity be prepared for it to not behave as expected. Sometimes infinity just goes bonkers and boom your single sphere is now a pair of spheres.

[@taleladar]

If sqrt(2pi) is your brilliant solution, then obviously something went horribly wrong.

[@kaanetsu1623]

I just love pi its literally everywhere

[@rodricrack1072]

I dont know if its ny ignorance or a fact, but werent the z=0 and z=1 the two poles of the analytic continuation?

[@eniky]

But you can't swap terms in an infinite series. It only applies to finite series

[@edomeindertsma6669]

Most of those things only work if the terms are finite, how can we now they also work for infinite ones?

[@xninja2369]

fun fact 52!>molecules present on earth so i think ∞! =√2π doesn't make sense though that's why I really hate and love math , i don't get point of this video like sum of all natural numbers is -1/12 (ramanujan paradox)

[@EdKolis]

This makes about as much sense as saying that the sum of all integers is -1/12. Wait, doesn't that involve a zeta function too?

[@gustavorc25]

Try extending the factorial to real numbers or complex and now combine this with the function Zeta.

[@cringotopia8850]

I like how he called the zeta function "infamous", it totally deserved it

[@programmingpillars6805]

its to totaly wrong if : infinity! = 1*2*3*4*5...... then infinity! > 3! with 3! = 6 > srqt(2Pi) where is infinity! > sqrt(2Pi)

[@ronaldtownsend5745]

This just goes to prove that if you want to break the rules you can make mathematics do anything you want. Foremost infinity is not a number. It is a concept. You cannot take the factorial of a concept.

[@garthreid355]

Bro how do you think about these stuff. It's mind blowing

[@BriTheMathGuy]

🤯

[@ryanmacinnes1930]

Pretty wild ideas, he might as well be making it up. Like ∞² = √π/∞.

[@adityapanchal534]

And if *infinity!* = √2π, then *(infinity!)²* = 2π ; which is ridiculous...

[@andrewolesen8773]

So many comments about not giving the warnings of analytical continuation, but we are just going to ignore the misuse of infinity?

[@jefdeleeuw]

Isn’t it really to say this is not true as ! is een increasing function and therefore it must that inf!>=10! >= sqrt(2pi) = inf! And thus I would be able to say that any x! Equals sqrt(2pi)

[@sinecurve9999]

This "proof" smells like using the geometric series to show that 1+2+3+4+...=-1/12. I can't ignore the factor of sqrt(2pi) in the denominator of the normal distribution and how the normal distribution is the limit of the binomial distribution for large n.

[@tomkerruish2982]

You can also use the zeta function to show it, since zeta(-1) = -1/12. I think the sqrt(2 pi) is more related to the Stirling approximation for n factorial (it's the "constant" term).

[@Noam_.Menashe]

How do you get from 1+2+3+4... To a geometric series? Honest question.

[@dlevi67]

@@tomkerruish2982 ζ(-1) diverges. Its analytical continuation via Dirichlet's η is -1/12. Another interesting observation (which can be justified and is connected to Stirling's approximation) is that the Gaussian integral value between -∞ and +∞ is also √π...

[@kjl3080]

Actually the exact same method is used for both!

[@jimschneider799]

Any idea if that's related to the ln(2 pi)/2 term in the Stirling approximation of the natural logarithm of the factorial?

[@vascomanteigas9433]

Yes

[@noah2123]

This needs context to make sense, infinity pie doesn't really equal root of 2 pie

[@Qermaq]

Engineer says: So it's just pi.

[@GaryFerrao]

Calm down people, this answer is only in the context of the analytic extended Riemann ζ Function.

[@GaryFerrao]

like how 0 ÷ 0 = 1 in the context of sin(x)÷x, but 0 ÷ 0 = 2 in the context of sin(2x)÷x.

[@BriTheMathGuy]

True!

[@ProfessionalRacist123]

Q: d/ds(1/n^s) = - sn^(s-1) for s != 1 and lns for s=1 0:50

[@tonymouannes]

While those formulas are way beyond my math level, I can tell you there is something wrong with the manipulation (ecen though I don't have the expertise to find where the mistake is). Factotials go up and for any number bigger than 1, it's going to he bigger than the original. So infinity with clearly give infinity. Your proof reminds me of the meme that proves that 1=2. You already said assuming we can use the properlize of ln for infinite sums (which sounds pretty suspect) and that you're using the other formula outside of its defined domaine. You're also stacking derivatives to the already bogus math. I feel that at that point someone can create whatever result they want. I've also seen some bogus math in the past done with infinite sum series of fractions that should be infinity but manipulated to give -1/12. I see something similar used at the end of your manipulation.

[@neuralwarp]

But we know inf! > n! for all finite n; and certainly for inf! > 3!

[@snejpu2508]

1^2+2^2+3^2+...=0, but it is infinity as well. So infinity factorial is just 0 factorial, which is 1. : )

[@ЯвсеРЕШУ]

Eu gosto disso! Boa explicação detalhada!

[@geektoys370]

how is that possiblle mate

[@jaimeafarah7445]

Can the result of √(2π) for ∞! be justified using the definition of factorial? that is assuming it could make sense if at all, then wildly would be equivalent to ∞! ≡ Limit n → ∞ { Γ(n+1) ₋ Γ(n+1, n) } Where the second term in the bracket is the upper incomplete gamma function.

[@thanhavictus]

Can you link the video in the description

[@danking9879]

This is what Aisac Euler would call ambitious

[@priyanshsuthar519]

The very step treats infinity as an integer. Well, no one could agree on whether infinity is surely some infinitely large integer, do they?

[@_Longwinded]

“If I were to awaken after having slept for thousand years, my first question would be: Has the Riemann hypothesis been proved?” -David Hilbert

[@douglasstrother6584]

If you can't keep it real, it's time to skate around the Complex Plane!

[@BriTheMathGuy]

⛸

[@thepeff]

Why does mathematics insist on making me afraid of the letter “s”?

[@Jack_Callcott_AU]

Something has gone wrong here. Clearly infinity factorial is not equal to any finite number. I think Knut Thompson has figured it out.

[@telnobynoyator_6183]

You can't really rearrange terms of an infinite sum like that...