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Ra Ra
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Comments by "Ra Ra" (@RaRa-eu9mw) on "ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12" video.

No. This is the universal value given to 1-1+1-1... Whenever it is given a value, the only one that makes sense and ends up being internally consistent is 1/2.
312
@efeyzee The geometric series test tells us that the series is divergent. If we're being religious about it and saying maths ends at Cauchy, we then have to indeed say "there is no value for this series." If we allow the rich development of analysis which came about after Cauchy though, then we find a whole slew of techniques showing us in multiple ways that 1+2+3+...=-1/12. This video shows one of those proofs. This is the only value which works for this sum, and gives us a consistent, useful answer.
77
@brandonklein1 I am a mathematician, and I say 1+2+3...=-1/12. I can find you millions of examples of other mathematicians saying the same. I even once attended a talk by Fields medalist Terry Tao about this topic and similar results. Is he somehow not "a mathematician in his right mind"? You are wrong when you try to separate a function and its analytic continuation like this. A function IS its analytic continuation. One entirely determines the other.
45
@cemrecevikoz This is the point, your understanding of infinity needs complete upheaval. It's not something we ever get nearer to. We cannot generalise finite results to the infinite case.
45
@nacholopez7348 No you can't?
38
@Skwertydogs That's a terrible argument and totally irrelevant. The result 1+2+3+...=-1/12 has myriad applications to the real world, via string theory, via L-functions, via class numbers...
35
As a mathematician, these results are completely correct. What do you say zeta(-1) is? How about H(0) Hurwitz number?
35
@nacholopez7348 Because the fact I'm a mathematician is not relevant. 1+2+3...=-1/12 is true whether or not any individual advocating for it is qualified. For instance, if we travel back to 1850 and work with the analysis of Cauchy and so on, then all mathematicians would say (as some commenters here are saying) that you can only assign values to convergent sequences, and anything divergent does not have a finite sum. I was responding to someone who was saying no mathematicians TODAY say that 1+2+3...=-1/12. That's wrong. All of us do. Literally every mathematician working in number theory has to use results like this on a daily basis.
33
@mikefochtman7164 It's you who violated a rule by bracketing in that fashion. You can't do that with divergent series.
30
@eternalio5885 Any mathematician which works with and studies this stuff absolutely believes that. If you're stuck with the maths of the 1800s then you aren't going to be able to make any progress in the study of L-functions (which are everywhere in analytic number theory) or class numbers (which are everywhere in algebraic number theory). If you say "1+2+3... cannot equal -1/12 it has to be infinity" then you are not someone with a different opinion, you would be denying such large areas of mathematical knowledge as to render you impossible to function as a researcher. So yes, mathematicians either do believe results like this, or they are working in fields like statistics where results like this aren't important and haven't learned about them yet. The last mathematician who studied this stuff and didn't believe 1+2+3...=-1/12 was Cauchy, who died in 1857.
22
The correct term for series is indeed equals. A sequence can approach something, but a series, when convergent, is defined as equal to the limit of the partial sums. As for divergent series, yes they are equal. We are assigning a value, exactly as we do with convergent series, in a manner which is useful and consistent.
21
@zoltankurti "everyone here talks about summation in the sense cauchy talked about it." There's the problem. Maths has come a long way since Cauchy.
18
@eternalio5885 A calculator cannot give any infinite series, only a mathematician can, and all mathematicians say 1+2+3...=-1/12
18
@NardKoning You are not allowed to permute the terms in a series like this unless the series is absolutely convergent. What you've demonstrated is an example of the Riemann series theorem, which is a much more beautiful result than you give it credit! It says that if what you just did was permitted we could actually get S to equal anything at all! Of course, in the video's proof of 1+2+3...=-1/12 this kind of rearrangement is never done.
17
@rogercline5377 I don't agree. In no context is it correct to say 1+1=45 without said context being given. However, 1+1 DOES equal something, and because we are the ones in charge we give that something a symbol: 2. You could give it a different symbol if you wanted, but this would be pointless. What you are doing, however, is the equivalent of saying "no, 1+1 doesn't equal anything. It can't equal anything. You can say you assign it the value 2, or rewrite it as 2, but you aren't allowed to say it equals 2." Of course, saying 1+1=2 is useful, and consistent, and meaningful, and so that's why we say it. Similarly, you are saying "no, 1-1+1-1... doesn't equal anything. You aren't allowed to say it equals anything." In actual fact, saying it equals 1/2 is useful, consistent, and meaningful. This is why mathematicians all agree that it equals 1/2. We are in charge, and we are steered by logical consistency and utility, and this result passes both tests.
13
@markwalden2433 He treats maths like a system of restrictions, and in particular he is treating Cauchy's results as laws which he isn't allowed to break. He says that any divergent series cannot be given a value, which... yes it can. We're in charge here, not Cauchy. If we can give a series a meaningful, consistent, and useful value then it has that value. This can be done for 1+2+3+4... There are many ways in which this series equals -1/12. He dismisses this problem by claiming, again wrongly, that the only link between the series and this value is the zeta function. Totally wrong. I'm a number theorist and the Hurwitz class number of 0 is -1/12. I can't speak to string theory but they mention a connection there too in this video. There are lots and lots of connections between 1+2+3+4... and -1/12. It's because these two expressions are equal.
13
@devalopr You're absolutely right. Infinity is not a number. A sequence cannot "converge to infinity" for example.
13
@snakeatwar Yes it is. The sequence of partial sums is 1,0,1,0,1,0... which obviously does not converge on a limit.
11
@zoltankurti This limit of partial sums definition is not obvious either, but is widely accepted throughout modern mathematics. Similarly, analytic continuation, smoothed summation, etc. these are all accepted and work. This video isn't supposed to be a rigorous exploration of those things. It's supposed to be a hint at the wonderful and unusual results which contemporary number theory has produced.
11
The maths here is correct. If you watch this video then you will have seen one of the many proofs that 1+2+3+...=-1/12 and one of the areas of physics in which this result is used (string theory). It is also used throughout other areas of physics and also pure mathematics.
10
@lievenvv The -1/12 result is, as mentioned in the video, used in string theory. It's also found elsewhere in physics. In terms of mathematics, it shows up in the meromorphic continuation of the Riemann zeta function and the Hurwitz class numbers - things of interest to analytic and algebraic number theorists respectively. Why not explicitly work with averages? Because the method of summation doesn't really offer us any insight. We could use this method to find 1+1: It is the limit of 1,3/2,5/3,7/4,9/5... (these are the averages of partial sums). This is of course ridiculous. Just say 1+1=2 and use that and move on. There are also methods for finding this result which do not use averages. For example, any form of integral or smoothed summation will find that 1-1+1-1...=1/2. We don't focus on the averages because the averages aren't important - ANY summation method which gives a value to that series will give the same value: 1/2
10
@seroujghazarian6343 It is not the limit. That sequence diverges, so has no finite limit.
10
@alshahriar6230 The key to remember is that strange things happen at infinity. There are some even odder results in this direction. For example, 1+4+9+16+... (all the square numbers) is 0.
10
@rogercline5377 You are confusing the partial sums with the series as a whole. As an example to help you understand this, consider the series 1/2+1/4+1/8+1/16... The series itself, you can hopefully see, is equal to 1, even though none of the partial sums equal 1. Indeed, none of the partial sums are integers, but the series itself is.
9
@seroujghazarian6343 Yes. You can't generalise finite case to infinite case though. This is one example of that.
9
@benaissa9684 There's a lot of mistakes in that comment. Most of this was cleared up by Cauchy. The limit of partial sums is ONLY available as a technique for convergent series. Divergent series, like 1-1+1-1... and 1+2+3+4... cannot use this definition. A sequence cannot "converge to infinity" as infinity is not a real number. If a sequence approaches infinity it does not converge to anything - it diverges. In your third paragraph, you are confusing "illogical" and "irrational" with "unintuitive." The result -1/12 goes against intuition, but it doesn't go against any logic. There is no logic saying "if something is true in the finite case it must be true in the infinite case." In the final paragraph, I agree. All mathematics must be logically consistent. However, its study and the definitions we use are driven by utility. It would be logically consistent to say "no series has a value, even convergent ones like 1+1/2+1/4..." This would not logically contradict anything (pre-Cauchy, many mathematicians held this view), but it would be useless. Whereas giving these series values, defining it as the limit of partial sums, is extremely useful, so that's what we go with. It is logically fine to say "1+2+3... doesn't equal anything. No divergent series have values" but it is useless. We go with the useful value of -1/12. Indeed it is the only useful value which is logical and consistent throughout mathematics.
9
Nimbo Stratus No. That's a completely different scenario. In the case of a quadratic equation, there are 2 answers which are valid solutions to the equation. A series is a single expression, and only has one valid, consistent value (or none at all). In the case of 1-1+1-1+... that value is 1/2.
8
@rogercline5377 You're forgetting that we're the ones in charge here. We get to decide the rules. There were people who opposed the use of irrational numbers, or complex numbers, out of some sense of cosmic law being broken - but statements involving these were so consistent and useful that they came to be accepted as mathematically true. If a series can be assigned a value which is consistent and useful then we say it is equal. There is nothing at all stopping us from doing so, and so, just like using irrational numbers and complex numbers, mathematicians now accept that 1-1+1-1... is 1/2.
8
@ToastFrench24 It diverges, yes. That doesn't mean we can't assign it a value. It just doesn't converge to one.
8
@benaissa9684 Logic provides constraints yes, but there are many choices and decisions in mathematics. In this case, -1/12 is not logically inconsistent with any other result, so it is fine to be the value. Application is, in many areas of mathematics, king. There are a million different ways one could define an elliptic curve, for example, and we settle on the definition which is most useful for generating other results.
8
@StephenWong14 The issue with what you've done there is that by ignoring the zeros and grouping the 2s together you are falling foul of Riemann's rearrangement theorem. This is not an issue with infinite sums so long as you ensure the resultant Dirichlet series still agree, which yours don't. In actual fact, your T does equal -1/2, which can be seen from any relevant dirichlet series, including (coincidentally) the zeta function. Hope this helps.
7
@findlaysa Let me know if I've misunderstood but I think you're picturing something like this: "Let Sn be the sum at n terms, so that S1=1, S2=3, S3=6 and so on. S1 is positive, and if Sn is positive then S(n+1)=Sn+n is positive. By induction, all Sn are positive" This reasoning is correct, but induction only lets us conclude something about the set being induced over. In this case, we've concluded Sn are all positive, but only for natural n. The sum "1+2+3..." could be reasonably given the notation Sinfinity but infinity is not a natural number, so the induction has not shown anything about this case.
7
@jazzabighits4473 No. A sum can only equal one thing. If it equals both 0 and 1 then you have that 0=1 which is nonsense. There is no such thing as "superpositions" of answers. The sum equals 1/2.
6
@brandonklein1 It is equivalence.
6
@luex6639 The series 1+1+1... can't be given a value by any summation method. You are also correct - it is divergent by various series tests. The series 1-1+1-1... is divergent by the alternating series test, but this doesn't mean it cannot have a meaningful value. That value is 1/2.
6
@rogercline5377 Yes. 1million - 1million + 1million - 1million.... is equal to 500,000. You can also see this by just factoring out 1million. 1-1+1-1+... is 1/2. Multiplied by 1million is 500,000.
6
@seroujghazarian6343 Lots of results are unintuitive or surprising when infinity is involved. It's a fact that 1+2+3...=-1/12. It's also a fact that this goes against what we would expect from the finite case.
6
@seroujghazarian6343 You are still making the same mistake - taking a rule for a finite case and applying it to infinite case. Take for example 1/2+1/4+1/8... All of the finite sums are not integers, but we cannot conclude from that that the series itself is not an integer - it is. The series equals 1.
6
@seroujghazarian6343 So is -1/12
6
@Tom Petitdidier Would you accept Terence Tao saying it? Or Hendrick Lenstra? Because they have.
6
@zoltankurti The sums of these series do exist, and are used by mathematicians like myself every day.
6
Nimbo Stratus Your post is nonsensical. I've read through it a few times but it's not at all clear what you are trying to say. Obviously you cannot simply take one of the partial sums. I never suggested you could. You can, with certain divergent series (or any convergent one) take the average of partial sums as n tends to infinity. Doing this will give consistent answers for convergent series, and also gives answers such as 1-1+1-1+...=1/2. It does not give any answer for 1+2+3+... For that you need some other method such as Ramanujan summation. Hope this helps.
5
@t00by00zer That would be Cauchy's answer, but mathematics has come a long way since Cauchy. We can find values for almost all divergent series at this point.
5
@lc9245 You could not be more wrong. Pure maths is now way more important in real world applications than applied.
5
This concept that youtube commenters have of "the traditional sense" needs to stop. Every context where the sum of the naturals appears, it is always taken to be equal to -1/12. It's useless talking about what the sum is "in everyday arithmetic" (whatever that is) when the sum never appears in everyday arithmetic.
5
@seroujghazarian6343 There is no such number as "positive infinity." Certainly, a series cannot "equal infinity." In this case, mathematicians are in agreement that the only sensible value to give 1+2+3... is -1/12.
5
@dpollak59 Yes it is. What is it not consistent with?
5
@fozymilograno What are you saying isn't true exactly?
5
@benaissa9684 Sure thing. Hurwitz class number of 0 is -1/12. Hurwitz class numbers are pure mathematics, a million miles from physics.
5
@NJ-wb1cz Uppsala :) And no, obviously not, but I enjoy mathematics and enjoy spending my time explaining mathematics to the general public.
5
We say that 1-1+1-1...=1/2 as this is the only sensible and consistent value to give this series. Every divergent summation method will give 1/2 (or will not be applicable to this series).
4
@jazzabighits4473 You are confusing partial sums which are finite with the value of this series which is an infinite sum. The partial sums are all 0 or 1, but the infinite series is equal to 1/2.
4
@nacholopez7348 With infinite sums, we are never using regular summation. You cannot literally add all the terms in the regular fashion. We are doing analysis, and the gradually built up tools of analysis do indeed show us that 1+2+3+...=-1/12 See this video for one example of such a proof.
4
@rogercline5377 I didn't say it converges. 1-1+1-1... is divergent. It still has a value though, and that value is 1/2.
4
@seroujghazarian6343 That's true for converging series only.
4
@benaissa9684 The series of partial sums diverges ("to infinity") yes, but a series is only defined as the limit of its partial sums when that sequence is convergent. You say -1/12 is inconsistent with infinity, and refer to this as the "most obvious result" but infinity isn't a result at all. Infinity isn't a number. No series "equals infinity", even divergent series. Prior to accepting the -1/12 value, we have no other result to compete with. It goes against intuition, sure, but many results do. Your final paragraph is a logical fallacy. We cannot take finite cases (finite sums) and generalise to infinite cases. There are myriad examples where such reasoning does not hold. In fact, the local-to-global principal is a tool which we have to prove each and every time we wish to use it. It is absolutely incorrect to say "this is true for a truncated, finite version so must be true for the infinite case."
4
@darkbrandon8431 1+2+3... absolutely is equal to -1/12. My field of expertise is mathematics, and this is not simply a quirky or unintuitive but useless result. Mathematicians use this and results like this every single day. It can be proven a number of different ways, and it is very useful.
4
@NJ-wb1cz There's no such thing as a "convergent value." A value is a value, it doesn't converge to anything. With regards methods for assigning values to divergent series, there are many. If you want a simple method, Cesaro summation you may find interesting. If you are happy with integrals, lots of forms of smoothed summation are available. If you know a bit more about complex analysis, many divergent series can be expressed as Dirichlet series which have analytic continuations and functional equations, giving the value. So yes, there are a large number of methods for assigning values to divergent series. What do they say about 1-1+1-1...? They all give it the value 1/2, because that's its value.
4
@requiredow7096 The "answer" is indeed -1/12. With regards making the rules, we do. There is no celestial diktat on mathematics. So long as our work is consistent, logical and useful we really can explore in any direction we want. What precisely is the issue you have with the reasoning presented in the video?
4
@steff2738 Yes it is. This is the point. You might as well say "there are no solutions to x^2=2, you can assign a value to it but it's not the same." In mathematics we find definitions which are useful and consistent and can be proven. This is one of them.
4
@zoltankurti I don't agree. It is entirely unreasonable to say the sum "approaches infinity" because it's a sum. It doesn't "approach" anything. It doesn't change value. It is fixed. With regards different summation methods, there are lots of different summation methods, and they all fit into two categories: 1) cannot give any value for S for one reason or another 2) gives the value -1/12 Because of this, there are only two valid answers to the question "what is the sum of all the natural numbers?" You can either say "I don't know" or "-1/12." If you say "it doesn't have a value" or "the value is infinity" then this isn't a different, equally valid response. It's just incorrect. No summation method says S doesn't have a value, and no summation method says S equals infinity.
4
Not quite. Think - the series 1+1+1+1... is divergent, but rearrangement obviously does nothing. The statement is really that any conditionally convergent series which isn't absolutely convergent can be rearranged to converge to any value.
4
@DionysiosArvanitakis And fields medallist Terry Tao? How about him?
3
@someaccount3438 Yes, the sum of all naturals equals -1/12. "It logically follows that since s is a sum of strictly positive terms, s must be positive" The above statement is precisely disproved with this counter-example. The key point being that finite cases cannot be generalised to infinite cases - the sum of finitely many positive numbers is a positive number, but the sum of infinitely many positive numbers may be a negative.
3
@lc9245 The fact that 1+2+3...=-1/12 is used extensively throughout pure maths, in L functions, in class numbers, in everything.
3
The video is correct though? I can rewrite every step in terms of Dirichlet series to show it all holds, and it would be more rigorous, but ultimately this proof is fine.
3
I mean, why wouldn't multiplication by two or shifting to the right be okay? We perform those algebraic manipulations all the time, with both convergent and divergent series. Nobody's going to arrest us for multiplying something by two.
3
@luke46219 I don't see where you are seeing a problem here?
3
@luke46219 Yes. 1+1+1+1... cannot be given any value. In the Zeta function, this would be Zeta(0) which is a (simple) pole. Summing two series which don't have a value together doesn't make sense, let alone summing infinitely many of them. Depending on how you do this nonsense summation, then, you can get any answer you want. Some of them will be correct, some of them will be just more nonsense. Watch: I'm going to add 1+1+1+1... to itself, but first I'm going to multiply it by -1 and shift it one space forward, so that I get 1+(1-1)+(1-1)+... = 1 But all I've done is add the series to -1 times itself, so this is 0. Therefore 1=0. Of course, I haven't really proven that 1=0. What I've done is started with nonsense and used it to construct more nonsense. Hope this helps? I may still be misunderstanding where you are seeing a problem here.
3
@NJ-wb1cz Because any technique which assigns the correct value to convergent series and assigns any value to 1+1-1+1... gives the value 1/2
3
@NJ-wb1cz Nothing to do with feeling superior or insecure. I'm just happy to explain mathematics. The more people thinking about and understanding maths, the better.
3
@kepe7323 Ah right. Analytic continuation refers to the method but also the result of that method. In the same way that "square root" is an operation, but "the square root of a number" is another number. The "analytic continuation of a function" is indeed a function, and all analytic functions are equal to their analytic continuations. Also, though it's not relevant to this particular discussion, analytic continuation extends the domain, not the radius of convergence. Analytic functions are not defined on domains containing points outside the radius of convergence.
3
S-S2=4S working term by term. S=1+2+3+4... and S2=1-2+3-4... so subtracting term by term we get S-S2=0+4+0+8... and 4+8+12+16...=4S Looking at the partial sums of S2 we don't get 1,-1,2,-2,3,-3.... but 1,-1,2,-1,3,-1....
3
You're making the mistake of thinking infinity can be treated as a finite case.
3
The issue is when you say 2 < 1 + 1 + 2 + 3 + 4 +... This is untrue.
3
Results from finite summation cannot be generalised like this to infinite cases.
3
What exactly is your issue with the video?
3
Divergent series can have values too (and most of them do).
3
They never claimed that the series are convergent. But divergent series with multiplicative values (such as these) can be given values via their L-functions.
3
@t00by00zer It isn't. This is one of many valid ways to show the result 1+2+3...=-1/12. They missed off one additional point (all these completed dirichlet series have same functional equation) but other than that, which I feel would have been too detailed, the video is perfect in presenting its proof.
2
@someaccount3438 We assign values to divergent series because it is useful to do so. In this video they talk about applications in string theory, but in maths there are applications in number theory. The Hilbert class numbers, for example, use these values in calculation. H(0)=-1/12 as it is the sum of the natural numbers.
2
@someaccount3438 It is a regular sum, and it is as objective as truth can be. I can understand that you want to follow intuition when it comes to infinity, but you cannot. This is just one of millions of examples where the finite is easy to work with and the infinite is very mysterious. It is never sound reasoning to say "because X is true for the finite case it must be true for the infinite case."
2
@someaccount3438 Kind of philosophical I guess? Outside of the context of mathematics, sums may not have any rigorous meaning at all!
2
@cb5542 The limit of the difference between the value and the sequence elements (or partial sums of the series) tends to 0. But, I'm not saying that the series is convergent. It isn't. I'm saying it is a divergent series which nevertheless has a unique, consistent, useful value.
2
@cb5542 "=" is perfectly well defined for how we use it. For divergent series it may be less intuitive than convergent series, but then convergent series are themselves less intuitive than finite sums. There's no issue in saying, for instance, that 1-1+1-1... = 1/2. This result is useful, meaningful, and consistent.
2
@cb5542 The value of a series is only defined as the limit of the sequence of partial sums when that sequence is convergent. In this case, it is divergent (as you rightly note), and so we use other methods, such as smoothed summation, to define the value as 1/2.
2
@thefinancialagent9608 It is divergent. A divergent sequence is any sequence which does not converge.
2
Because shifting it to the right means adding 0. This has no effect on a sum.
2
You are pretending maths stopped at Cauchy. It didn't. What you have posted represents our knowledge circa 1850. This video is where knowledge is at today.
2
The series 1-1+1-1...=1/2. The fact it's divergent doesn't change that. There are many ways to find the values of divergent series. Cesaro summation, integral summation, smoothed summation... They all find that 1-1+1-1...=1/2.
2
@MuffinsAPlenty This is really just the perfect example of double standards. You are being unnecessarily aggressive towards me, characterising me as having simply said "no you're wrong" when in fact the original comment here is, pretty much word for word, saying nothing more than "no this is wrong."
2
@seroujghazarian6343 Correct, but we still say 1+2+3... = -1/12
2
@mrmomonk616 What in particular do you disagree with? Happy to explain in more depth.
2
@David-cc8lz The proof in the video is basically fine. All of the series featured can be written as Dirichlet series where the manipulations hold. They didn't mention this, but doing so would have required a much more in depth video than they clearly wanted. Anyone familiar with the mathematics understands that there's a couple of details going unsaid, and anyone unfamiliar with the mathematics doesn't really need all those details in a youtube video.
2
@curtishung9897 The issue with this is the regrouping you perform on the second/third line. This can only be done on absolutely convergent series.
2
You can assign values to divergent series. This isn't the 1850s any more. We assign values to divergent series all the time.
2
1+2+3...=-1/12 is honestly a wonderful, surprising result that mathematicians and physicists use every day. It's right to be on numerphile.
2
@nich8244 The analytic continuation of the Riemann zeta function is one way to get it, but not the only way. You can go via a number of different Dirichlet series, or you don't have to use Dirichlet series at all. This video's proof is equally valid.
2
@rakshitya9826 "You can't assign it one value", sure you can. And yes, the value of 1/2 is correct.
2
@sejr8053 S2 isn't infinity, it's 1/4.
2
@sejr8053 What are you saying no to exactly? I think the video shows perfectly well that S2 does not "equal infinity."
2
@NJ-wb1cz I am referring to the evaluation of convergent series as correct. For example, any method which correctly finds 1/2+1/4+1/8... = 1 and is generalisable to 1-1+1-1... will give it the value 1/2. Any method, which finds the indisputably correct value for convergent series, finds 1/2 for this. Your suggested alternative, of "just picking 1/10" is not a method which gives us any testable results for convergent series. 0/0 is undefined, you cannot divide by 0.
2
@NJ-wb1cz I'm not following you. There are no methods which give any value for 0/0.
2
@NJ-wb1cz I'm a number theorist.
2
@icandothisallday986 I'm happy to discuss any bit you don't follow or have doubts about. Please do say!
2
@kepe7323 A function is indeed its analytic continuation. If you have some analytic function, and tell me the values of the function on any compact set (no matter how small) then I can work out the analytic continuation from those values, and what I recover is identical to the function you had in the first place. This is what's meant by "a function is its analytic continuation." They are identical.
2
@terranceparsons5185 0.9 recurring is the same as 1.
2
@terranceparsons5185 0.9 recurring is indeed equal to 1. There are many ways to demonstrate this, and you might want to read other mathematicians on the topic rather than youtube comments. A common proof is that 1/3 = 0.3 recurring, and so multiplying both sides by 3 we get 3/3 = 1 = 0.9 recurring. Hope this helps.
2
@Anthony Hernandez Assigning values to infinite series is something we've been doing since the 1800s. It's well established, useful, logical mathematics. It's also frequently intuitive - If I break a chocolate bar in half and give you half, and then break my half in half and give you half of that, and continue doing this, you know intuitively that the end result of me doing that "forever" is that you would get the whole bar. The concept of infinity is a tricky one, but we can absolutely do lots with it. It's just not a number. You can't talk about something "equalling infinity" or "3 times infinity" or nonsense like that, but you can talk about sequences tending to infinity, or the values of infinite series.
2
@Shivian124 Yes it can be. There's no law stopping us from giving divergent series values, and the only logically consistent option for 1+2+3..., used throughout mathematics, is -1/12.
2
@Octavian999 It's not really an assumption. The Dirichlet series 1-1+1-1... has an Euler product expansion which continues analytically to 1/2.
2
@Octavian999 Divergent is not the same thing as having no value. Divergent series with multiplicative entries (as is the case with all series here) almost always have some value or other.
2
@gigantopithecus8254 Yes. The series is divergent. This doesn't mean it doesn't have a value, or we cannot find its value.
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@tirterra1222 I don't follow you - the cesaro sum is the limit of a cesaro convergent series. It's the same exact value as summation by any other method. For example, if we consider 1/2+1/4+1/8... we get 1. We get 1 if we take the limit of partial sums, and we get 1 if we take the cesaro sum. Two different methods for arriving at the same value - the value of the series itself. With regards your second objection - yes, you cannot rearrange divergent series. But the video doesn't do any rearrangement, so there's no problem there. Your final objection is just a bit odd - the video doesn't even mention the Taylor expansion of the series, or use it in any form.
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@tirterra1222 This video does none of the things you mention. It's just manipulations of Dirichlet series.
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The vast majority of divergent series can still be summed. S1 does indeed equal a half.
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@NotBroihon No mathematical rules have been broken here. Indeed, as you freely admit, numerous different summation methods find the same result: 1+2+3...=-1/12
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They didn't break the Riemann series theorem at any stage. The video is legit.
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@John Doe I am a mathematician, an algebraic number theorist, and I use results like this all the time. With regards the sum 0+1+2+3... yes this equals 5/12. You say this is "obviously absurd" but the fact you find something surprising isn't an argument at all. Lots of results in mathematics, especially ones incorporating infinity in some way, are surprising.
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@John Doe Then you're making a philosophical point rather than a mathematical one. People reacted in a similar way when Cauchy sorted out convergent series, saying that what he was doing "wasn't summation" but "something else." The fact is that the results were logically valid, consistent and useful, so mathematicians ultimately concluded that yes, this is the same as summation. Similarly, most mathematicians today say yes, this is summation. 1+2+3... really does equal -1/12.
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Then you'll die on the wrong side of the debate. This result is used in many aspects of maths and physics every day.
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There isn't one. You have to take the infinite sum to get this. Similarly, there is no smallest partial sum where 1/2+1/4+1/8+... is a whole number, but the infinite sum is just 1.
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Yes, and using ramanujan summation I get that 1+1+5+1+9+1+13+1+... = -1/3 which agrees with the series shown.
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What makes you think divergent series don't have values? We use the values of divergent series all the time.
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Because the result you stated is only true for finite sums, not infinite series. A similar law would be "The sum of two fractions, in simplest terms, with different denominators, is never an integer" Consider the series 1/2+1/4+1/8... Our law proves that none of the partial sums are integers, but it doesn't disprove that the entire series is an integer (it equals 1).
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It is true. Myself and many other mathematicians use results like this every day.
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I'm afraid it is true, and many mathematicians and physicists use this result every day.
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What do you mean when you say "the standard definition"? If you want to ignore all mathematics after Cauchy, and take his discoveries as the limit of mathematical understanding, then yes, 1+2+3... is a divergent series so cannot be summed. But a lot has happened since Cauchy. Today, the result 1+2+3...=-1/12 is undeniable. It's used all the time in pure and applied mathematics. It is very much "the standard definition."
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1-2+3-4... doesn't flip between 1 and -1.
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@douggwyn9656 Important point - "by rearranging terms." Yes, if you have a divergent series you are not permitted to rearrange terms, but they do not break this rule in the video.
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It's not. This result is widely discussed and used in many areas of mathematics and physics. You can't "debunk" that.
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@MuffinsAPlenty The result in this video, and the method used to achieve it, are totally legitimate.
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A divergent series can still have a value. 1-1+1-1... diverges, but it equals 1/2.
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@zoltankurti Any summation technique which gives a value works, for instance ramanujan summation.
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@Gunno77 People said the same thing about Cauchy's work with convergent series! That it was different concept and not right to call it summation. The fact is that we're in charge here. We get to decide what is called summation, and it is most useful and tidy to define it in the most generalised way possible. As a result, summation includes adding integers, and adding complex numbers, even though the latter were not known about for thousands of years. Cauchy gave us methods for summing convergent series, even though this was out of bounds for thousands of years (and many mathematicians considered it impossible). It seems you are happy up to this point, but not going any further. Dirichlet, Riemann, Cesaro and Ramanujan, among many others, gave us methods for summing divergent series, as well as alternative methods for summing convergent series. Importantly, all these methods agree on known results from finite sums or convergent series. You are essentially saying that our understanding of summation is allowed to grow and evolve up to about 1840, but must stop there and we're not allowed to use anything that came since (or must "call it something else"). This is illogical, and needless.
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@Gunno77 I don't mean to patronise at all! I am here to educate, and I think anyone engaging on here will be intelligent and fully able to grasp the concepts. Apologies for coming off as patronising. I don't know how often you read mathematical papers, as the thing you say you've never seen happen happens all the time. You couldn't really get very far in automorphic forms, L-functions, class groups, or related topics without routinely utilising summation methods for divergent series. If ever I see a value given for a divergent series, I won't think "which method did they use?" as any will do! When you talk about it "not being a sum" but just "assigning a value", this is semantics. People said the same thing about Cauchy's work on convergent series. The thing to remember is that we are in charge. We get to say that 1+1/2+1/4+1/8... really does "equal" 2, and really is a valid sum, even though nobody could perform it on an abacus. Our mathematical knowledge and understanding evolves and grows over time. It used to be that we had no method for performing the above summation, but thanks to Cauchy, now we do. It was subsequently the case that we had no method for performing the summation of divergent series, but thanks to the work of many great mathematicians, now we do.
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@Hyperion1722 You absolutely can, and results like this are used throughout mathematics. Number theory, group theory, and mathematical physics regularly use results like 1+2+3...=-1/12.
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Why exactly do you think it is false? Results like this are true and used throughout modern mathematics and physics.
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@Octavian999 I don't really see the relevance of the objection. Ramanujan summation is indeed one of many techniques for finding the values of divergent series, but it doesn't "redefine how summation works" by any stretch, and also this video doesn't even use Ramanujan summation. It's using (under the hood) the analytic continuation of various Dirichlet series. Notice how every expression is multiplicative? That's not a coincidence! When you say "you can't just say that S1 is 1/2" I guess my response is "who's going to stop me"! Mathematicians have taken Grandi's series to equal 1/2 since the 17th century, if not longer. Wherever 1-1+1-1... appears today, it would always be understood to equal 1/2. I can't think of any context at all in which it would have any other value, or in which a mathematician would simply walk away and declare it unknowable (as seems to be your preference?)
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@bedhead-tb4qg No. A divergent sum is simply defined as a sum which is not convergent. There is no such thing a sum which is neither divergent nor convergent.
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It's not misinformation. It's true.
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@shadow-ht5gk I am a number theorist, so I hope I know something about this topic! What in the video do you disagree with?
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@MuffinsAPlenty There's no need to become abusive. If you have a problem with modern number theory, group theory, physics, then voice that. It's unlikely entire branches of academia are going to agree with you, though.
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@bradwavemb What exactly do you think is wrong in the video?
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@bradwavemb Hi! Sure. What exactly do you think is wrong in the video?
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@bradwavemb All the series here diverge, but I don't think it's right to say they are manipulated "as if convergent." Convergent series are just one of many things which can be manipulated in the ways shown here. Finite sums, obviously, can also be handled like this. Importantly, though, is that all the steps here are valid for L-functions. Divergent series can be written as L-functions whenever the terms are multiplicative, which is the case for every series here - everything is, under the hood, an Euler product, and so all the steps are valid. I don't see how the video "misleads" or "deceives" the audience. They state outright that they don't want to get into the weeds with Euler products and L-functions and Dirichlet series and so on, and I think that's the right call. The justification as to why each step is valid is way too high level for a popular mathematics video like this. All the audience needs to know is that the steps are indeed valid (which they are) and that the result is interesting (which it definitely is).
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It's correct. There are no flaws or mistakes in this reasoning, and the end result is used in many areas of pure and applied mathematics.
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You're trying to make a joke, but have actually stumbled upon saying something correct. In complex analysis, we do indeed (when needed) take the value of sin at infinity to be 0.
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@Arthur Rusteberg Yes, I think what you are getting at here is correct. A function is its analytic continuation. It's like looking at an elephant's trunk and saying "what does this look like 3 meters back?" The answer is "a tail" because what we're looking at, in full generality, is an elephant. Those who treat Cauchy as dogma would say "it doesn't look like anything 3 meters back, there's no trunk back there!"
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@kepe7323 The square root of a number is the same as the number if and only if the number is 1 or 0. The square root of a function (as in, the composition of square root with f) is the same as f if the image of f is a subset of {0,1}. In other cases, the square root will not equal the function itself. The analytic continuation of f, on the other hand, is the same as f if f is meromorphic, which all Dirichlet series are. As for the second question, there is no such thing as the "domain of convergence". The two concepts of "domain" and "radius of convergence" are very different. The domain of a function is the set of points on which it has been defined. The radius of convergence is a concept which applies to power series, not all functions, and is the radius of the maximal disc where the series converges. I can change the domain of a function easily. For instance, if f is constant 0 and the domain is positive real numbers, then there is a very obvious way to extend this domain to all real numbers. Note that this doesn't change the function - it's still constant 0 - but extended to a bigger domain. The radius of convergence, on the other hand, is set in stone. A power series has a radius of convergence and this cannot be changed without changing the function.
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@tirterra1222 I can't talk about the string theory side of things. I'm a mathematician and can only address the mathematical side. In terms of mathematics, this result is everywhere. It crops up all the time in Number Theory and Group Theory in particular.
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What's the average of two 1s? How about three 1s? Or fifty 1s? It doesn't matter how many terms we add, the average is still 1. The limit of an average as the number of terms approaches infinity is not necessarily 0.
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Not quite. You've proven that the partial sum for any finite n>2 is greater than 3, not that the sum of the series itself is greater than 3. Similarly, if you consider 1/2+1/4+1/8... the partial sum up to the n-th term is less than 1, but the value of the series itself is not.
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Is this a joke? You know the mathematics behind this underlies all of modern cryptography right? You know that internet shopping and banking could not exist securely without the understanding of the mathematics at play here? This is far from worthless.
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@danielcohen227 It is absolutely possible. In fact, it is true, so it must be possible! You are misunderstanding the nature of infinity - it is a very peculiar thing and unusual results like this are common.
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Myself and many others in mathematics accept that this result is correct. I am happy to discuss it more if you'd like.
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@pedrolauand9605 Not much to say to this because you didn't really prove or demonstrate anything, just made assertions which are incorrect. 1+2+3...=-1/12 is indeed seen in the zeta function, but also other contexts. It is used in various branches of physics but also in number theory (Hurwitz class numbers). It is simply a fact that 1+2+3...=-1/12 and this video has just one way of proving it.
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@pedrolauand9605 So why is H(0)=-1/12 again? Feel free to take your time dancing around the fact that 1+2+3...=-1/12. On level of maths, I'm a postdoc in number theory, so feel free to use any terms you want in your argument.
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@pedrolauand9605 Why do you think H(0)=-1/12? Is it random? Or is it that for a very specific reason?
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@pedrolauand9605 If you are studying Cauchy sequences, then yes in that context we say divergent series do not have defined values. We also say 1/0 is undefined but as soon as you enter the hyperbolic setting we say 1/0=i infinity. In every context where we use 1+2+3... we say it equals -1/12
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@MuffinsAPlenty I'm struggling to think of an example where what they did here would be an issue with any divergent function. They explicitly say they're not going to go into detail with analytic continuation etc, but this would work for any L-function and thus any divergent series, no?
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@John Doe That's Riemann's series theorem, and demonstrates why you cannot rearrange or regroup terms in divergent series. This was not done in the video though. Everything done in the video is completely fine.
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@MuffinsAPlenty I don't think it's a big problem that the audience were not given the full details. As you rightly say, the audience here likely doesn't understand L-functions, so the video was absolutely right to leave them out.
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@IsomerSoma What specific issue do you have with what I wrote? I'm more than happy to clarify.
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@davidk7212 With infinite series, you often get unintuitive results like this. You cannot generalise what you know from finite sums to infinite sums. For example, 1/2+1/4+1/8+1/16... is never an integer, but at infinity is just 1.
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@douggwyn9656 Three applications from physics alone is already impressive, and that's not even touching on the applications in mathematics. "Whether it has legitimate uses is debatable." It is absolutely not debatable. Study of L-functions and class numbers are just about the most important parts of contemporary number theory, and both use results like these extensively.
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I don't know where you read that but it's false. Infinity plus anything is still infinity.
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Many divergent series can be given a consistent, logical value. If you accept the values of convergent series but not the values of divergent series, you are saying that the only valid mathematics is the mathematics of 1860. Everything up to Cauchy is correct (1+1/2+1/4... =2) but everything Cesaro and beyond is incorrect (1-1+1-1...=1/2). This is a strange position to adopt.
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positive+positive=positive only applies for finite sums.
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@MuffinsAPlenty Yes, it applies for convergent series, but not for series in general.
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S2 isn't infinity, it's 1/4
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Hi. I am not involved with the video, but am a mathematician, so feel qualified to answer your points. 1 - Yes, we must be. Of course. There are many examples of finite results not generalising to the infinite case. In mathematics this is called "failing the local-global principle." In the video, I believe they exercised sufficient care. However, some of your subsequent arguments do not show this care. 2 - The sum of 1-1+1-1 is defined for finite, or infinite number of terms. You can see, for example by Cesaro summation, that the infinite series is equal to 1/2, as stated in the video. 3 - A sum cannot tend to anything. The sequence of partial sums can, and in this case the sequence of partial sums tends to infinity. However, this doesn't tell us anything about the value of the infinite series, because that value only equals the limit of the sequence of partial sums in cases where the sequence is convergent. For example, 1+1/2+1/4+1/8... has the sequence of partial sums 1, 3/2, 7/4, 15/8... which converges to 2, and so the series equals 2. In this case, the sequence of partial sums diverges, and so we need to use other methods (such as the method in this video) to find the value of the series. The second paragraph reiterates that the sequence of partial sums tends to infinity. This is again an attempt to use a local-global principle where one does not exist. Your quoted formula is correct, but only correct for finite n, and there are many cases where we cannot generalise such a result to the infinite case, of which this is one. With regards the use of an equal sign, this is the correct sign to use, because the two things really are equal. 1+2+3+4... has a value, and it is -1/12. In mathematical writing, if I wanted to say that one element was "associated" with another I would probably use the tilde symbol (~) and define what that meant in context. There are conventions for certain common equivalences. For example, we can associate 6 with 11 by the fact they have the same remainder after dividing by 5, and this would be written with three horizontal lines, as 6≡11. Of course, the most common "association" is equality, where we use the = symbol, as in this case. Finally, I can't respond to your integral argument as that's something I've never come across before.
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It is math. Mathematicians use and obtain results like these every day.
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@aadithyahrudhay2269 the sequence of partial sums doesn't converge to anything, but the divergent series absolutely has a value.
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@aadithyahrudhay2269 The sequence of partial sums (the nth triangle number) doesn't have a limit. The series, though, does have a value: -1/12.
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@aadithyahrudhay2269 Ramanujan summation is one example of smoothed summation. It all uses integration sadly, but it should be intriguing to you that all the methods of summation do one of two things: 1) cannot find a value for 1+2+3... 2) find the value -1/12 The issue is that a LOT of strange things happen at infinity. If you are going to work with infinity at all you are going to find these bizarre, counterintuitive results. I think of them as beautiful, and the more surprising they are the more beautiful.
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@aadithyahrudhay2269 I'm an algebraic number theorist, working with p-adic L-functions, so I literally use this result every day! Look up the Hurwitz class number of 0, and find out why it is the value it is, and you will see a lot of what I study in the process.
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This is just by definition. It's how we define the value of such a divergent series. Similar to how 1+1/2+1/4+1/8... is 2, because we define it to the limit of the sequence of partial sums.
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Yes. This result can be arrived at several different ways.
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The average of the partial sums is equal to the sum of the series for all convergent series.
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Focux Nope. 1+2+3...=-1/12. Myself and many other mathematicians use results like this every day.
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Add the series 1-1+1-1+1... to itself? Either you get 2-2+2-2+2-2.... or 1. Either way, you don't get the original series back again.
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@zoltankurti Both.
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@SteveXVII The issue is that your manipulations break Riemann's series theorem. You are essentially doing (1-1)+(2-1)+(3-1)+.... and then rearranging to (1+2+3...)-(1+1+1...) which Riemann's series theorem tells us we cannot do. Further, you can only do Series1-Series2 when both these values are well defined. In your case, Series2 is not defined. You say it "equals infinity" and then say "anything minus infinity equals -infinity" but this is nonsense. Infinity is not a number. Hope this helps.
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@Gunno77 Not at all. Summation means the same thing as it always did. Cesaro summation is just one valid way of finding the value of a sum with less strict requirements than many other methods. You can find the value of a sum by adding up the numbers one by one. Or, for certain sums you can use things like the arithmetic or geometric progression formulae. For convergent infinite series you can use Cauchy's methods. For some (but not all) divergent series you can use Cesaro. For some you can use integral methods or analytic continuation, and so on. This is all still "summation." Just different ways of arriving at the same correct answers.
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@blackscrow 1+2+3+4...=-1/12 is true entirely in the normal sense, and via conventional algebra. There is no special conditions or context for this statement - it is true full stop.
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@blackscrow What they did here is correct. We can perform various manipulations on divergent series (though we cannot rearrange elements, as in absolutely convergent series) and their manipulations here were valid.
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@hdjdco5428 You are confusing the finite and infinite cases. Results from finite cases cannot be generalised to the infinite case.
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