Does 1+2+3+...=-1/12?
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- Опубліковано 28 вер 2024
- The answer to the question in the title is that it depends on how you define the infinite sum function. Until the infinite sum function is specified, the question is not well-defined; it's the equivalent of asking "what is f(5)?" without defining the function f.
The sum of all natural numbers, and the sums of divergent series more generally, are often debated because it seems incoherent to say that a divergent sum equals a finite number. This video explains the machinery used to obtain finite outputs from divergent infinite series.
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Music: C418 - Pr Department
Excelent explanation. Usualy, people disagree on awnsers, not because the resulut is different, but because the interpretation of the question, or the definitions of the terms, is different for each one.
yes. quite right. who said: “Fraternal correction requires a scrupulous audience.” ?
I think what frustrates people is that we expect numbers to represent real-world things. Imaginary numbers are sort of an exception to that, but one is fairly easy to tolerate because they at least factor into results that are clearly in nature (for example the imaginary character of sine waves). But this "-1/12" business ... ? It doesn't really manifest in the real world, and that makes it feel like trickery.
Except, it turns out there is something known as the Casimir Effect, where the -1/12 legitimately factors in. I won't claim to understand it, but it does exist as a real-world thing.
I Love how deep you think about problems!
The *equality* and it's meaning was great (and also when you referred to Integral).
Thank you so much dear *Mu Prime*
Grande, las integrales son la mejor herramienta para estos menesteres, ya que las sumas nos pueden dar diferentes resultados, bprp tiene dos vídeos al respecto con diferentes resultados.
Y un resultado de la sumatoria de Ramanujan usando integrales con el resultado de - 1 / 12. Esta también se puede avalar con Regulaciones en un vídeo de MathArg Papers. Y con otra integral una tercera demostración de SingingBanana.
Deep thinking simply presented 👍!
Brilliant as always!
🙂👻
Very nice explanation, thank you
Pretty sure math is just broken... Kind of in the same way speed runners break games, mathematicians can break math.
A good video, but I still think it's wrong to tell people that 1+2+3+....=-1/12 because it is just obviously false, it just bewilders people to be told that, and what is the point of that. The truth is, as mu prime says, that a zeta function is defined for all complex z with real part of z >1 and that this function can be extended by analytic continuation with Re (z)
Similarly 1+2+3+4..=1+(1+1)+(1+2)+(1+3)..=1×inf+1+2+3+4=>S=S+inf=>inf=0 😂
"Obviously false". Today in 2023, we still hear this. In a math video, after math and science education, and a seemingly functionning brain, we can hear the "argument" "obvisouly false". After all that humans went through; under an excellent math video, with clear explanation, we can still hear " obvisously false".
I bet you make fun of platists, don't you?
It's really worth mentioning that zeta(-1)=-1/12 but not infinite sum of natural numbers.
Doing that would literally miss the point.
You obtain -1/12 only if you define the infinite sum with the continuation of the Rieman Zeta function. In fact, following the definition you chose for the infinite sum, you can obtain whatever result you want. That's why I always consider that this is pure cheating. The serie does not converge. You can say the limit goes to infinity, or has no value in R.
Mathematics is also about notations ...
I can surely accept writing F(1,2,3,4,5,6,...)=-1/12 , if you give a precise definition of F (like zeta continuation), but certainly not 1+2+3+4+5+6+... = -1/12 which is somewhere something provocative.
"You obtain π²/6 only if you define the infinite sum with the limit of partial sums" so i consider it to be pure cheating...
@@manun7105 Sorry ? who's talking about π²/6 ? The infinite sum 1+2+3+4+... just has no value, because it is not convergent.
I have a question. If I wanted to sum 1+1/2+1/3+1/4... in the same way. Is that also equal to -1/12 or is it something else?
It turns out that 1+1/2+1/3+1/4... can't be evaluated in the way that 1+2+3+4... can using the Riemann zeta function! In analysis terms, the analytic continuation of the Riemann zeta function has a pole at s=1.
@@MuPrimeMath oh, thank you for the heads up. I appreciate it. 🙏
@@neiloppa2620 There is a certain regularization that gives you that 1 + 1/2 + 1/3 + ••• = γ, the Euler-Mascheroni constant, but it works differently than the Ramanujan summation that gives you 1 + 2 + 3 + ••• = -1/12. The Ramanujan summation tells you that 1 + 1/2 + 1/3 + ••• is undefined.
0:32 - 0:35 Strictly speaking, "infinite sums" are not sums at all. We can only ever rigorously interpret them as some type of function being applied to a sequence, and that function will output a real number or complex number.
0:39 - 0:46 Yes! I am highlighting this, because this is a point many people fail to understand.
1:06 - 1:17 This, with the caveat that we acknowledge that + is an associative operation. Otherwise, this would not work. This is why power towers are difficult to work with for people. The notation suggests associativity, even though there is none, resulting in ambiguity.
1:47 - 2:23 Yes, and this truly is what lies at the center of controversy. Almost everyone can agree that what we are doing is applying some kind of function, a function that takes a sequence as its input, and gives a real or complex number as its output. The question is, what is this function? That really is what the disagreement is about.
2:58 - 3:31 I know this definition is the first definition of a series anyone is ever introduced to, and it probably is also the most basic. That much I acknowledge, and of course, if any calculus student is reading this, then let me just say that, at least in your calculus course, this is the definition you will be using. However, recognizing this is different than saying this is the standard definition. To say it is the standard is to imply that it is the definition used in almost all contexts, and that any other definition is obscure by comparison. This is not really the case. This definition gets introduced before any other, if only because you use this definition to work with asymptotic expansions: particularly, Taylor expansion and Laurent expansions, in real and complex analysis. These are definitely extremely important and have many applications. Outside that, though, this definition sees very little use, if only because of how shortsighted it is. And when I say that, I am not talking about using Cèsaro summation or Ramanujan summation in replacement. No, forget those things. Even if we keep the discussion focused to only summing sequences where things are nice and convergent, this definition is still problematic because it cannot be generalized in a particularly satisfactory way. It only works if you are trying to sum functions defined over the integers. In many areas of mathematics, and even sometimes in physics, though, you want to sum over functions defined over other infinite sets, such as the rational numbers. This definition will not do in those situations. Besides, the definition relies on a rather naive definition of infinity, where the distinction between countable and uncountable sets is not made. This is important when extending summation to integration, where the measure theoretic formalism is used. The general definition that gets used, even in the context where we limit ourselves to convergent sequences, is somewhat more complicated than the basic definition, but is also in more widespread use than the basic definition. So advertising the basic definition as the standard is misleading at best, in my opinion. Perhaps I am being nitpicky, and maybe this is more an issue of semantics than I am reading into, but that is how I see it.
5:26 - 5:41 Another way to concisely put what Mu Prime just said here is that these divergent summation functions are extensions of the basic summation function you get introduced to when you learn calculus. For an analogy, Ramanujan summation is to Cauchy summation (that is the technical name of the basic definition) as the Gamma function is to the factorial function for natural numbers. Writing something like (-1/2)! = sqrt(π) is technically an abuse of notation, but is fully "correct in spirit," and it communicates the point across rather well. 1 + 2 + 3 + ••• = -1/12 is also an abuse of notation, and some consider it a far more egregious abuse of notation, but again, it is just as "correct in spirit." At the end of the day, if we are being objective here, all this is telling us is something about the extension of Cauchy summation. And that is really what this is about. At its deepest, when you think about, this entire controversy is basically akin to having a debate about whether writing something like (-1/2)! = sqrt(π) or 0^(1 + i) = 0 is acceptable or not. It is less a debate about mathematical concepts, and more a debate about how we should use notation to express an idea. Because, here is the thing: no one who writes 1 + 2 + 3 + ••• = -1/12 is insisting that lim n·(n + 1)/2 (n -> ♾) = -1/12. What they are insisting is that the notation in question is valid for their purposes.
5:48 - 5:54 Well, technically, the definition used is that of the Cauchy integral. But it just so happens that for bounded continuous functions, the Riemann integral exists and is equal to the Cauchy integral if the Cauchy integral exists. So, eh, minor details, I guess.
6:53 - 7:30 Yup, and that is the best summary of the controversy I have seen any UA-camr give on the subject. I say, excellently done.
Thank u! It’s really cool explanation) 🥺🥰
Masterfully done!
It can be viewed just fine as a question of equality; you're just choosing a different equivalence relation on the space of sequences over your set. Video would definitely be improved by mentioning some of the alternative modes of summation convergence, minimally including Cesaro, Abel, and maybe Euler summation.