the most famous Ramanujan sum 1+2+3+...=-1/12

What is the sum of all the natural numbers? Isn't it just infinity or is it really -1/12? Here we will see 1+2+3+... "will be" -1/12 if you use the Ramanujan summation, which is a way to assign a value to a divergent series.
Check out Bose Integral: • Zeta function in terms...
Check out NOT -1/12, • Not -1/12
Sum of 1/n^2 by a Gucci Integral: • a spectacular solution...
Sum of n^2, • how Ramanujan did 1^2+...
Ramanujan Summation: en.wikipedia.o...
Ramanujan summation in detail: hal.univ-coted...
The sum of all natural numbers:
by 3b1b: • But what is the Rieman...
by Numberphile: • ASTOUNDING: 1 + 2 + 3 ...
by Mathologer: • Ramanujan: Making sens...
🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
💪 Get my math notes by becoming a patron: / blackpenredpen
#math #mathforfun #blackpenredpen #calculus #ramanujan

[@Nondas8552]

When you have to prove that the sum of all natural numbers equals -1/12 at 10 pm and go on a date at 11 pm

[@blackpenredpen]

Yup!! : )

[@leif1075]

@@blackpenredpen isnt it only true if you include inaginary numbers like negative one square root..otherwise it is impossible and incorrect! Because the sum,always increases..

@@leif1075 If you taking the sum of all counting numbers from one to infinity then your sum is already increasing continuously. What controversy on this integral sum!!

[@johnny_eth]

It's not equals. The sum is divergent. The Ramanujan summation is a transform that assigns values to divergent series. Kind like how a Fourier transform gives you the domains of frequencies and phases of a periodic signal but it NOT equal to the signal.

[@19divide53]

@@johnny_eth The equal in OP's comment is obviously in a Ramanujan summation sense

[@fCauneau]

That's what I call a Christmas gift ! Thanks !!! And Merry Xmas to you and your readers !!

[@ayushrathore9190]

Watching this at 2:45am. Any freaking maths lover like me?

[@masrock9203]

ayush rathore I’m watching this at 4:20

[@jafethmendezgomez691]

I am watching this at 3:20PM :)

[@ofgokce]

02:30. 15 minutes before you!

[@adsumquifeci6015]

2 a.m. in Poland.

[@darwinschuppan8624]

2:31 for me

[@TheKing-cn2ou]

my common sense just refuses this answer

[@GoreGutztheImpaler]

Good cause your common sense is right. It's more complicated then this but if you change the definition of + to something else it will change what 2+2 =

[@stevemenegaz9824]

Ramanujan included this result without explanation in his first letter to GH Hardy. Hardy put the letter down initially as rubbish. We are exceedingly lucky that Hardy reconsidered the letter and didnt toss it in the trash like the 2 other British mathematicians that recieved the letter.

[@Ididntplayball]

I’m shocked. The maths I loved were just the very basic.

[@rogerkearns8094]

black props red jacket. (If 'props' include hair and spectacles.)

[@badmanjones179]

black pen red suit

[@David-ww2sg]

*tom scott has joined the chat*

[@zwz.zdenek]

Still YAY!

[@gabbarisback6052]

😂😂😂

[@mukeshkumar_manigilla]

Legends thought

[@alanturingtesla]

Square root of -1‚ now convergent-divergent series. In a few years, I expect dividing by 0.

[@tricky778]

@Alan Turing, which size of 0? a big 0 or a little 0? I'm sure if you divide by a big 0 you get a smaller result than a little 0

[@arnavanand8037]

I feel like for that we need to redefine division. Just like there's gamma function for factorials

[@GynxShinx]

lim(x->0+) 1/x->infinity If you're too lazy to write the whole thing then 1/0+=infinity

[@arnavanand8037]

@@GynxShinx I think everyone knows the limit already. But people feel unsure about the actual answer

[@oracle7858]

Eagle Shows Down 1/0+ is infinity but what is 1/0 🤔

[@themeeman]

Looking sharp

[@blackpenredpen]

Clingfilm Productions Thank you!!!!

[@radiotv624]

This is fascinating, I love Ramanujan

[@kingbeauregard]

Me too, especially the chicken-flavored stuff.

[@davidrheault7896]

The monster is back again...i know it from physics waves in vacuum. -1/12 casimir effect

[@radiotv624]

David Rheault That’s right! What is/was your major

[@davidrheault7896]

@@radiotv624 I did a specialisation in physics It is above major.

[@indicgamer2907]

Ramanujan's work opened a new world of mathematics that astrophysicist use to study black hole , time travel , free energy quantum tunneling

[@nekososu]

or you should say the zeta(-1) is -1/12

[@abhiruppaul5601]

correct bro

[@arnoldo-probjeto3111]

Not correct bro. In fact, ACzeta(-1)= -1/12, where ACzeta is the analytic continuation of zeta function, that is NOT the zeta function.

[@19divide53]

@@arnoldo-probjeto3111 But we call the analytic continuation of zeta function by ζ(s) as well. A bit of abuse of notation but I think that's the convention. In fact, in Riemann's 1859 paper he did denote the analytic continuation of zeta function (sum of n^(-s) over all positive integers n) ζ(s), which you can see by looking at the paper online. There's no reason to say ζ(s) may only be the sum of n^(-s) over all positive integers n. When I'm doing some other problem unrelated to the Riemann Zeta function, I could define a function, say, s^4sin(3s)-cos^5(s^2)+6/s-9/s^7, and call that ζ(s). That's just as valid as calling my function f(s) or g(s).

[@moumous87]

2:00 you got to love this guy for putting the R on top of the = and for actually showing what the heck is this "Ramanujan summation" thing. Thank you

[@saxbend]

Mathologer wants a word.

[@NateROCKS112]

Mathologer actually hinted at what is going on in this video. He showed that the area under the x-axis of f(n) = n(n+1)/2 is -1/12, which can be interpreted as a definite integral from -1 to 0 of n(n+1)/2. n(n+1)/2 is a graph of this infinite series (as n→∞) since it's the sum of all natural numbers up until n. Also, BPRP used the proper notation with the Ramanujan summation, and did not claim that the natural numbers "summed" to -1/12, just that they could be assigned to it.

[@angelmendez-rivera351]

NateROCKS112 Mathologer did show this, but he equally also spoke against any association between adding the natural numbers and obtaining the result -/12 via analytic methods. The problem lies in how poorly and inconsistently the technical terminology of calculus is being used, and how poorly the word “summation” is interpreted. The problem is that, nowadays, when dealing with sequences of partial sums, we tend to associate them to summations themselves, without much justification, even though these summations are not sequences, but numbers, and the summations are furthermore the result from a set of rules of arithmetic operations on a field, whereas statements about these sequences are statements about vector spaces and linear functionals. When we see the sentence 2 + 2 = 4, this statement is provable. This has nothing to do with sequences. It is a statement that is the consequence of the rules of addition and basic arithmetic. When we see that f(x) = x^2, and we calculate f(1) + f(2) + ••• + f(100), this is also a statement about arithmetic, and summation notation is merely a notation to abbreviate this result: the operation is ultimately still addition. I see no reason why 1 + 1/4 + 1/9 + ••• = π^2/6 should be treated any different. In fact, to prove this identity, we typically just use algebra and trigonometry, not much from calculus itself though. I see no reason why 1 + 2 + 3 + ••• = -/12 should be treated any different either. Caught hated the concept of infinity. He was, so to speak, an ultra finitist. This is why he invented the concept of a limit, and whenever we wanted to speak of adding infinitely many numbers together, he rejected the idea, and instead proposed to talk about the limit of the sequence of partial sums. For some practical purposes, and for his specific theoretical, both mathematical and philosophical purposes, this definition works just fine. Nowadays, we treat it is a method to assign value to infinite sums, but strictly speaking this method does not give you the sum itself. In fact, as I already said, Cauchy would have said such sums do not exist, and the expressions are nonsensical, precisely because he did not believe in the notion of infinity. So associating actually adding infinitely many terms with these limits is conceptually incorrect, and an equivocation of bounded behavior with infinite quantity, which should not happen. What the limit of the sequence of partial sums tell us is information about the asymptotic behavior of some function, and this function represents an algorithm. In the case of the natural numbers, what this limit does is answer the question, “What happens when, at every step of some process, I add the next natural number to the total I already have?” It tells us that this process will result in a number which at every step is larger than it was before, and at an increasing rate. I am not succeeding in getting closer to some value when I do this process. This is the proper meaning of what divergence is for this case. This tells us nothing about ACTUALLY adding the natural numbers, which intuitively and arithmetically should have nothing to do with sequences (because if we defined addition by sequences, it could never be commutative or associative, but we know addition is a commutative operator for fields). In fact, it tells us nothing about adding infinitely many numbers in general in the first place. We make the association because we want to and because it is practical in some contexts, but again, they strictly are not the same thing, especially when you understand the mathematical logic behind the axioms and definitions. Obviously, mathematicians tend to understand this, but when students learn calculus, they do not learn any of the rigorous details behind the definitions or theorems involved, so naturally they get confused.

[@Lolwutdesu9000]

@@angelmendez-rivera351 this is a brilliant post.

[@literatedouchebag]

@@angelmendez-rivera351 i couldn't have said it any better. Amazing post my dude

[@angelmendez-rivera351]

VeryEvilPettingZoo “But it’s the only definition that makes any conceptual wrong.” And that’s obviously wrong, as mathematicians from the 17th century have proven with their understanding of divergent series, which literally preceded any understanding of limits of sequences. The fact that surreal numbers and transfinite addition exists, without need to use limits and all, further proves this point. You must know any mathematics beyond those of your calculus II textbook if you think it’s the only conceptually sensical definition. But the other reason you are wrong is because there is no such a thing as “the ordinary conceptual understanding” of something in mathematics. That doesn’t exist. Mathematics is strictly and exclusively about what axioms you work with. And depending on the axioms, something is true, or it isn’t. And here is the catch: there is no standard set of axioms that is used in every field of research. Even within the same field of research, if we are having a conversation and I decide to get a different level of rigor, even there we’ve already changed the scope of the axioms and of the theory. Because of this, there is no ordinary understanding of anything: any understanding of anything just comes from axioms and that’s that. Math has nothing to do with intuition, and because of this, nothing is ordinary or conventional. That is what people refuse to understand. The only reason some things seem conventional to some people is because they’ve been limited to only specific math courses or they were exposed to something first for the longest time before being taught something they personally would consider “non classical”. If you had been raised with a calculus course learning about Ramanujan summation first and then being exposed to it continuously to work with it in engineering for the rest of your life, then you would perceive that to be the only conceptually ordinary sense to add infinitely many things. But that too would be an illusion. If there were such a thing as an ordinary conceptual understanding, then it wouldn’t be necessary to learn maths in such a way that in every new course, everything you learned previously is a lie and that there is another way to look at things. And look: mathematicians said the exact same thing about complex numbers centuries ago. The fact that people continuously make these claims and centuries later are always disproven is an obvious indication to the fact that such ideas about understanding are subjective and merely illusions, artifacts of tradition, not real in any mathematical or logical sense. “...of what it would mean to add up the terms of that series “forever”.” There is no such a thing as adding things up “forever” in mathematics. I don’t care if you put it in quotation marks or not: that word should not be in that sentence, not even in a remote, metaphorical sense. You’re projecting feelings and intuition that don’t exist in mathematics into the subject. Adding infinitely many numbers does not have anything to do with time. If I add one by one, then yes it would take me a very long time to finish the process, mechanically speaking, but processes don’t define anything in mathematics. If we can agree that a sum of infinitely many numbers can have a value, then that has nothing to do with time. Talking about time only furthers the misconception about what infinite sums represent notionally. If I have a way of adding infinitely many numbers and knowing what that is, then in principle there is no reason I shouldn’t be able to do this immediately, in less than a picosecond, or faster. And if time was of relevance, then it would be impossible to talk about infinity in the first place, so the concepts of convergence and divergence would make no sense whatsoever. “Conceptually, it’s indisputable that adding up the series “indefinitely” drives the sum up towards infinity, not -1.” And once again you made the mistake I pointed out in my comment. It’s like you just missed my point altogether. If you have a sequence defined by steps, where in every step, you add the next natural number, then yes, you get a process by which this number will increase to infinity as more and steps are performed, arbitrarily. And unfortunately for your argument, that’s NOT what the sum itself is. Summation is an operator, not a process. Nothing in mathematics in any field is a process. The only field in mathematics that talks about things related to processes may be some subfield of discrete mathematics concerning computing power and what not. But those things don’t define operations, nor should they. A mathematical identity and the process you get by it computationally are strictly unrelated. Perhaps children have a strong association between operations and processes, because it’s the only exposure they’ve ever had of operations, since they cannot really understand the abstract essence of what operations are, but that’s it. If I can add all the powers of 2 at once, then there is no “driving the sum” anywhere because there is no process, there is no sequence. Any extension of an operation should capture this abstract essence of what the operation is rather than any false non/existing notion of a procedure. You can choose to create a mathematics which is based on procedures, maybe define something call procedure theory. But that wouldn’t be arithmetic of real numbers anymore, that would be, well, procedure theory. And worse for the argument is that this concept of limit to infinity depends strictly on the type of infinity you are choosing to use as a boundary condition on the real line. Calculus on the projective line looks very different. Which only proves my point further. I will watch the video, though I hardly doubt they’ll say anything fruitful I haven’t already addressed, unless they end up agreeing with me. I say this as I’ve been in over 50 different discussions about the subject.

[@jamesnapier3802]

The infinite series 1+2+3+... never converges to -1/12. Claims to the contrary are simply false.

[@stratonikisporcia8630]

I does not converge, it diverges to an infinite number that can be represented as -1/12 but is not EQUAL to -1/12

[@YourPhysicsSimulator]

So elegant... I'm talking about you, though.

[@blackpenredpen]

Thanks!!

[@punambanik5310]

@@blackpenredpen sir, can you please tell me the name of the formula you used to find the summation??

[@spudhead169]

Thank you, both Numberphile and Mathologer didn't really explain this properly, that it's a "non-canon" summation.

[@rykehuss3435]

Nice way of saying its just made up by Ramanujan to goof around with

[@martind2520]

This is awesome. Can you do a video proving that Ramanujan Summation is consistent with regular summation for non-divergent series?

[@MrLecancre]

Try with cv riemann series

[@angelmendez-rivera351]

The discrepancy between saying that the series diverges and the sum is -/12 is that they are not even talking about the same thing. The fact that it diverges is a statement about asymptotic behavior and sequences, whereas the latter is a statement about arithmetic and infinite sets. Calculus is a theory about sets and functions. When we deal with summations in calculus, we never truly deal with an infinite summation, so to speak, although we do call them this out of bad tradition (just like how calling imaginary numbers imaginary is bad tradition). In calculus, what we do deal with instead is sequences of partial sums. Why? Because this tells us about the algorithm of after one number after another. If I start adding the natural numbers, in a specific order, and form a sequence for every step, then what is the behavior as I increase the number of steps? The behavior is that this sequence simply becomes infinite, every number I get is larger and larger on several orders. Although we tend to associate this with adding infinitely many terms, this is not what is truly happening and is merely an informality that happens in the calculus classroom, since explaining the real details behind standard analysis is complicated and outside the scope of the syllabus. In a sense, using the limit of the sequence of partial sums is already a way of assigning values to infinite series. It already is “a summation method”, but it is an error to call it a summation in the arithmetic sense because it obviously is not, since it has very different properties. This bears no contradiction with what Ramanujan postulated because Ramanujan is not talking about sequences and their limits. What Ramanujan is trying to do instead is imitate more closely the properties of summation as given by number arithmetic. Naturally, since we are adding elements of infinite sets, the results we produce are counterintuitive and outside of what induction can allow us to prove. Keep in mind, though, that using Ramanujan summation is not the only way to arrive at this result. Abel summation and Borel summation are both more intuitive than Ramanujan summation, and they also give -/12 as a result. Also, here is an interesting article by Terrence Tao that justifies this non-classical result in a very convincing way. terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/#zeta-s One thing I find interesting is that the integral from -1 to 0 of x^2/2 + x/2 is 1/6 - 1/4 = 2/12 - 3/12 = -1/12. It is interesting because this integral is the area between the x-intercepts of the function f(x) = x^2/2 + x/2 = x/2•(x + 1), which is equal to the nth partial sum of the sequence of natural numbers for x = n.

[@user-ft2vp5yw6p]

Wow, just wow. Amazing

[@noname_whatsoever]

Very well written. This clarifies what seems contradictory when the topic is usually discussed without a proper perspective. Thank you.

[@DanNguyen-oc3xr]

"Naturally...counterintuitive." Damn it.

[@suprafluid3661]

@@DanNguyen-oc3xr But i was going to say that hmmm.. 😣

[@abustefano8225]

Mmmm... Something wrong here. According to Abel-Olana formula en.m.wikipedia.org/wiki/Abel%E2%80%93Plana_formula the series diverges. Please elucidate

[@dayzimlich]

One of your best videos yet - keep up the great positive attitude in 2019!

[@blackpenredpen]

dayzimlich thanks!!! : )

[@federicopagano6590]

we have 2 options depending on which class we are in. But wich one is the correct answer if u are alone no classroom and u have to answer

[@blackpenredpen]

That will be depending if I want to make my life easy or not.

[@federicopagano6590]

blackpenredpen OK got the point diverges lol

[@angelmendez-rivera351]

Federico Espil He did not say that. Obviously you want him to say that, though, because you already are prejudiced and already believed beforehand the sum diverges. I wrote a comment addressing this.

[@shoobadoo123]

When applied in the real world, -1/12 is the answer. This appears in physics, string theory specifically. You can google to learn more there’s a lot of info!

[@benjaminbrat3922]

Thank you very much for this much-needed relativism when talking about divergent series. You could go even further by presenting several alternative summations (Cesàro, VP, Lambert, Borel, etc), I know Cesàro to be quite easy to grasp, and this would contribute bringing diverging sums enough breathing room for actual exploring, instead of bantering. Happy New Year!

[@djsmeguk]

What would be interesting is how the ramanujan summation formula is derived.. does it concur with convergent series as well?

[@angelmendez-rivera351]

djsmeguk I believe the answer is yes, but I’m not 100% certain.

[@nathanisbored]

i dont think its derived, its just a definition

[@angelmendez-rivera351]

nathanisbored It is inspired by some derivation. The definition only plays in a role in the domain of functions to which it applies. It is a function of functions.

[@angelmendez-rivera351]

Markus Steiner “I think it's neither derived nor it's just arbitrary.” Be more specific with what is it that you mean when using the word “derived”. This may not have a derivation from the axioms, but it has a rigorous foundation on time-scale calculus, which is a theory of which the calculus of sequences is a special case. It also has equally rigorous foundation in complex functional analysis and set theory. It uses the projective extension of the complex plane, known as the topological Riemann sphere. Also, it is not arbitrary, because it can be shown that, given any argument from heuristics, the continuation entailed by the heuristics is formalized with this method presented in the video. If any continuation to summation should exist so that divergent series could be evaluated, then this must be the one. Namely, had the method of convergence of the sequence of partial sum never existed, the field axioms of addition would heuristically entail these results. “The thing is that it also has to work for not divergent series.” It does. In his following video, which sums all the square numbers of the natural numbers, I wrote in the comments section that it is easy to prove that this series sums every convergent geometric series to its correct value, and it appears as though it does so with series involving trigonometric functions as well, as well as for the sum of all 1/n^2 for n in N. “So there might be more possible formulas which achieve this with different results for divergent series.” There are. Borel summation exists, as well as Abel summation and stronger linear methods. Wheel algebra and transfinite set theory can also derive these results independently. “Choosing one of these possibilities is in fact arbitrary.” No, it is not. If two different theories derive the same conclusions, then this entails the theories are, at the very least, consistent, but it could be the case that one is a sub-theory of the other, or even more radically, that given some super-theory containing both theories, that they are equivalent theories. Being able to prove the law of cosines using either Classical Greek geometry, or else using a Hamiltonian algebra of 4-ions does not mean that choosing either theory as the foundation for a method to obtain results is arbitrary. That is not how math works, obviously, nor has it ever been this way.

[@sl.murarikumar9675]

kzbin.info/www/bejne/pZXRZ6qbgN-eZ7s Ohhhhhhh Nooooooo Ramanujan questions answer is right or wrong Very confusing -1/12. OR. -1/8 🤔🤔🤔

[@blazedinfernape886]

The views should be -1/12

[@kishanthakor971]

😂😂yaaa

[@gabbarisback6052]

😂😂

[@shivamchouhan5077]

🤣

your comment is at 69 likes already, so I'm using *this* comment as a like button instead

[@pardeepgarg2640]

@ I ruined that 69 likes 😈😈

[@unironicaluser1867]

i understood until about 0:17

[@jzanimates2352]

You should make a collab video with 3blue1brown or numberphile!

[@europeankid98]

YES

[@late7245]

1-1+1-1+........=1/2 confirmed

[@hamsterdam1942]

Nope

[@xevlonperc]

No, it's 0.

[@late7245]

(-1)^inf = 0 so (1-(-1)^inf)/(1-(-1)) = 1/2 like Ramanujan

[@angelmendez-rivera351]

Xevlon Perç How would it be 0?

[@angelmendez-rivera351]

Георгий Шередеко The results of the video imply the equation.

[@beatoriche7301]

Great video! If I may ask, how do you actually derive this formula?

[@angelmendez-rivera351]

UPDATE: I found that if one takes the polynomial in n of degree s + 1 that gives you the sum 1^s + 2^s + ••• + n^s, then if we integrate the same polynomial in x with respect to x from -1 to 0, one obtains the same value one would obtain if one applied Ramanujan summation instead. The key is in using Faulhaber’s formula, finding the antiderivative of it in x, where n = x, then set x = -1, and the resulting formula is equal to -B(s + 1)/(s + 1), where B(s) is the s-th Bernoulli number. This gives the same result as Ramanujan summation, which also gives the analytic continuation of ζ(s). I’m mentioning it because I thought it was interesting. So this generalizes the result from my first comment.

[@submarino006]

I like your funny words, magic man

[@matthiashannesson7239]

-1/12th?

[@elthomaso10]

This is hands down the most terrifying video title I've ever seen.

[@biswadeepghosh5568]

Great presentation, felt so good since this concept is related to the work of Ramanujan, many thanks to help me understand this concept.

[@kimothefungenuis]

Better than numberphile

[@b77vedantmore51]

Now. seeing this video.... 'Yes -1/12 ' next video 'not-1/12'

[@jjeherrera]

Nice as always. Still, it would be interesting to understand what is Ramanujan doing that helps him to make the analytic continuation.

[@science-y9209]

Ramanujan is dead but he was a genius..

[@sanatankaushik456]

This is the power of the "Man who knew Infinity"!

[@adamwalker8777]

If I take 1,2,3... coins from every inhabitant of the multiverse, then I will remain in debt.

[@jevaughnclarke6174]

Maybe these sums have a greater meaning and we are not yet smart enough to know their meaning.

[@moiqtheplayer]

Something's wrong with Ramanujan's summation, although you made excellent explanation, I still disagree due to some invalid maths that Ramanujan did, but I don't think you guys realize it, do you?

[@ulilulable]

My reaction at 3:00 "what the actual... what the ... what‽"

[@SebastienPatriote]

This just went next level!!

[@ianprado1488]

How many markers do you buy a month?

[@pianochannel100]

Its the characteristic number of the summation, im not sure that saying the sum is actually equal to -1/12 is right

[@jorgecanales4864]

Si lo tuvieras tus videos con sub titulos en español, serian aun mas geniales, eres un genio

[@Podzhagitel]

has aprendido inglés?

[@1dash133]

How can a series that approaches infinity resolve into a simple fraction? How can a series of positive numbers resolve into a negative number? Sorry, I don't buy the validity of this mathematical modeling. EDIT: In researching this subject further, I see that the problem with this video is a failure to identify the mathematical frame of reference. It is something called "Analytic Continuation". It is a logical extrapolation that makes sense within its respective frame of reference. Sort of like discussions of division by 0. The playground being discussed is not conventional math.

[@mikahamari6420]

Look at Mathologer's video debunking this. This is total bs.

[@CosmicGoat-d3o]

I don't get it. adding positive numbers should be a positive number right?

[@NotBroihon]

Actually don't watch the numberphile video. It's probably one of the worst videos on Brady's channel (no hate). Instead watch mathologer and 3b1b :)

[@guilhermefreire8093]

Can someone please explain me how he got to that first formula??? It makes zero sense to me

[@JonJonFtheW]

The Ramanujan summation formula? He didn't "get" to it, he just wrote it down. The Wikipedia page screenshot is how you'd arrive at the formula.

[@jjeherrera]

The truth is we ignore how he got many of his results, even though people have been able to come up with proper demonstrations. He was an exceptional genius.

[@adrienanderson7439]

Sometimes you have to make an assumption that something is true in order to work with it, for instance I can take the derivative of something like x^3+3x^2+5 ( which is 3x^2+6x ), using the power rule, but i might not explain why the power rule works because its explained already somewhere else(en.wikipedia.org/wiki/Power_rule) and it would make finding derivatives take a long time to have to explain every time you do an operation thats maybe not + - / * or ^.

[@guilhermefreire8093]

@@adrienanderson7439 no this is different, because i assume that most of the audience does not understand ramanujan summation. More important than the derivation would be understanding just how a divergent series can be assigned a finite value.

[@jangjerdum6523]

2:10 what the f*ck is going on, Chen?

[@youtubecommentsguy9805]

Only step I could follow him on was when it went from -1/(6*2) = -1/12 🤣 e:// after watching again I also understand -it-(-it) = 2it 😎

[@minipashki]

and what about -1/8? S = 1 + (2+3+4) + (5+6+7) + (8+9+10) + ... = 1 + 9 +18 + 27 + ... = 1 + 9*(1+2+3+...) = 1 + 9S 9S - S = -1 S = -1/8 and by the way it's a minimum value of a function y = x(x+1)/2, which gives a sum of x terms of S

[@minipashki]

ah, ok - next video

[@Jacob-uy8ox]

Ramanujan must have been very high when he came up with that formula xd

[@gergodenes6360]

Not that high, because it can be used in many ways, it's just not a good way to represent things like this, as it should only be used on non-divergent series in the first place.

[@Jacob-uy8ox]

@@gergodenes6360 exactly, and this sum is of course divergent

[@angelmendez-rivera351]

Gergő Dénes What? No, the formula is a DEFINITION, and it was defined for divergent series. The formula was not made for convergent series.

[@angelmendez-rivera351]

Jacobo Zapata There is a lot of complex analysis that goes into the formula. Maybe you should do your own research before trash talking a great mathematician.

[@Jacob-uy8ox]

@@angelmendez-rivera351 I'm not trash talking about Snrivasa Ramanujan, in fact he is one of my favorite mathematician and I know the usage of that formula in complex analysis, I just wanted to do a little joke about the formula itself

[@scarbotheblacksheep9520]

If "diverges" and -1/12 are correct depending on how you think of it, could the domains of these be different? Like, is the Ramanujan summation including transfinite numbers in its sum, or something like that?

[@wysciguwka7231]

-1/8

[@youtubecommentsguy9805]

That whole video was the kind of vibe it was in high school when you cannot follow a single step what the math teacher is teaching you understand nothing but don‘t dare to ask because you feel like he does a good job and is a genuine friendly guy so you don‘t want to interrupt him.

[@Harlequin_3141]

I would highly recommend the Mathologer video over the Numberphile one. His is much more informative, the Numberphile one will lead a novice very far astray with some dubious math.

[@iclutchyt1002]

When you realise that 1+2 is 3 And 1+2+3+4+... is -1/12

[@harshitkumar4760]

It means that negative numbers are greater than positive and smaller too, it means its a cycle which repeats positive after negative and again positive. Think it this way as we increase the angle in tangent function its value reaches to infinity and then negative! And then again positive.

[@rainbow-cl4rk]

Now calculate zeta(s) with ramunajan summation

[@fmakofmako]

Would it be possible for you to do a video on analytic continuation or smoothed sums? Both have relevance to this video and the riemann zeta function.

[@samueldeandrade8535]

I am not the biggest fan of blackpenredpen, but I have to say that from the several youtubers I saw talking about this, he was the most respectiful, intelligent and interesting. Simply amazing.

[@blackpenredpen]

Thanks!

[@maccook9951]

At about 3 minutes in I had to actually check this wasn't uploaded April 1st

[@joryjones6808]

But I don”t understand that last bit. I’m in calc 1 and can see that integral from -1 to 1 of (1/x) = ln |1| - ln |-1| = ln(1) - ln(1) = 0 why would it diverge?

[@martind2520]

One does not simply integrate 1/x from -1 to Mordor.

[@angelmendez-rivera351]

Martin D LMAO

[@Austin_916]

I love math 😛

[@anweshaguha7366]

Okay, so I'm in love with that "Hmm" before you start solving a tough integral, I mean tough for me😊

[@meta04]

3:03 Why _wasn't_ I expecting you to have played that...

[@mockingbird_proxi]

The integrant looks also like the bose-einstein statistic🤔 And the f(t)-f(-t) term is the degeneracy g(t) of the energy state of the partical, in other words of ~t (the integration variable). Hmm, if f is even g is 0 and if f is odd g is 2f. So the integral is something like the pratical density in a way of summation😅 Now I'm confused

[@karlhannoush]

7:11 this is called the balls integral

[@VivekYadav-ds8oz]

Bro you getting married or something? EDIT: Completely irrelevant to what I said above, but I lost it at 3:03 with those little kids aaaaing 😂

[@blackpenredpen]

: )

[@adamoksiuta4715]

I think this is all wrong, because sum of positive numbers HAS TO BE a positive number, so it hasn't be -1/12, -1/8 or something else with minus before the number.

[@NotBroihon]

No. This sum is not equal to -1/12 and bprp doesn't claim it was. This technique "assigns" a value to a divergent series to classify it.

[@divergentmaths]

@@NotBroihon -1/12 is the correct (intrinsic) sum (regarded as "remainder modulo infinity") for the divergent series 1+2+3+4+5+6+...

[@stratonikisporcia8630]

Does it have anything to do with the p-adics such as the 10-adic form of -1/12 = ...3,333.25

[@Gideon_Judges6]

So could this "R equals" be replaced with a triple equals, meaning is defined to be equal?

[@akk92278]

Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined. ...

[@adamoksiuta4715]

For me it doesn't make sense. How sum of positive numbers can be negative? It's impossible.

[@Smertopia]

Let me prove 1+2+3+4 etc. So we start by writing down f(xz)/2*acb(blabla) ??? Profit. How does all that come from a simple som of numbers?

[@mikejackson19828]

Can you make the sum of the natural numbers to be whatever you want it to be?

[@tianvlasic]

The sum can also be 0; S=1+2+3+4+5... 2S=2+4+6+8+10... -S=1+3+5+7+9+11... 3S=1+1+1+1+1... S=1+2+3+4+5+6... 2S=0-1-2-3-4-5-6-7... 2S=-S S=0 So now you are saying that’s -1/12, i say it it 0, and logic says it is infinity. So it can’t be 3 values, possibly more. I am not saying that Ramanujan isn’t right i am just saying that it is a paradox. A=2+4+6+8+10... 2A=4+8+12+16+20 -A=2+4+6+8+10... -A=A A=0 B=1+3+5+7+11... 2B=2+6+10+14+22... -B=1+3+5+7+9+11. -B=B B=0 S=1+2+3+4+5... S=A+B S=0+0 S=0 S=1+2+3+4+5... S/2=1/2+1+3/2+2+5/2... S/2=(1/2+3/2+5/2...)+(1+2+3+4+5...) S/2=1/2(1+3+5...)+(1+2+3+4+5...) Now you see S/2 is bigger than to S. Of course -1/12 isn’t the answer. Since we proved 1+3+5... is 0 we can have a relation; S/2 is equal to S This satisfies -inf, 0 and inf so it means it is that three values at the same time. So you see the confusion.

[@AliBarisa]

The Ramanujan summation does not seem to make sense. Unless that sum of -1/12 does not exist in the real world. Since, The first three terms of the integer series( 1+2+3+...) have a sum of 6 and the sum gets bigger with each extra term. So, -1/12 cannot be the real sum. Unless we move to another "world" akin to complex world for "√-1"

[@kipgoin111]

i somehow found this ... wrong. With geometric progression, we can obviously prove the sum 1 + 2+ 3 + ... is greater than 0 ( while 0 > -1/12 => sum 1 + 2 + 3 + ... is not equal to -1/12 )

[@vansf3433]

If you denote the continuous sequence of additions and subtraction as S, then S will stand for an endless or undefinable series of such arithmetic operations, but not a single and well-defined quantity. Hence, the result of the arithmetic operation as a single and well-defined quantity, after such nonsensical manipulations of human -invented defective notions of mathematics, and the endless series S are obviously 2 different values. An endless series of arithmetic operations, no matter whatever sort of arthrimatic operation it is, it will never ever stop to give any result So, An endless series of numbers and a number are two complete different mathematical notions. One is undefinable and the other is definable Therefore, once you have claimed that they are equal, you have also unconsciously claimed that you have been out of your mind. No need of any proof here If s = 1 + 2 + 3 + 4 +... Going to infinity 4s = 4 + 8 + ... going to infinity could be possible, then it would be the same as claiming that that S = 4S, 1 = 4, 2= 8 , 3 =16, 4 =32 ... because the length from the starting point of the 2 endless series to infinity is the same or constant, and thus the number of additions of the first endless series and the number of additions of the 2nd endless series have to be the same for the length of the first endless series and that of the 2nd series to be the same because both of the endless series of additions start from the same starting point and stop at infinity, which is the same ending point Here are a 1-D visualisation of the 2 endless series of additions, which show that their lengths of addition operations have to be equal, while their numbers can be different C = 1--------------------------------------------infinity 4C= 4--------------------------------------------infinity It is obviously nonsense What's wrong with such nonsensical manipulation here is that the guy took advantage of the ambiguity of the notion of going to infinity, which is obviously an unknowable unknown or undefined value, to insert this baseless claim. But the 2:series of additions have to have the same number of additions because they both begin from the same starting point and ends at the same ending point. The ending point here is infinity, and there can be only one infinity, but not 2 infinites. Hence, the number of additions of the 1st endless series of additions and the number of additions of the 2nd endless series of additions have to be the same. Both of the resultant outcomes of the 2 endless series of additions have to be infinity, having to stop at infinity, because infinity is the largest possible value, although they have different terms in the same number of additions. So, his mathematical statement is the same as claiming that 1=4, 2 =8, 3=16, 4 =32 because that is the only one way for the 2 endless series of the same number of additions to end up with the same final output, which has to be infinity, but that only one way is obviously totally absolutely absurd Ignorant human mathematicians have the same problem of misunderstanding of the actual roles of human-invented numbers, and considering the unknowable unknown or undefinable value "infinity" as a specific value in their nonsensical mathematical claims Logical reasoning can always give you correct understanding all sort of human-invented notions, but not ignorantly and blindly fanatical applications of such absurd notions The only one thing which such nonsensical mathematical manipulations can prove is that human-invented notions of mathematics are defective with numerous flaws, and that is why and how you can prove such nonsense as an unknowable unknown result to be equal to a specific value

[@rykehuss3435]

This is the biggest clickbait equation of all math. Obviously its not true under real arithmetic, and involves some very high level math trickery and made up rules to get it. When people see arithmetic looking equations, thats what they think. I can say "1 = 2" , but under the hood I've changed the rules of math and logic to get that and you'd never know. I can still claim to be right because I used Rykehuss summation.

[@klerulo]

IMO, this summation of a diverging integral should just be considered a hash function for it .

[@bluesparrow-hx5qf]

The sum diverges to infinity, no matter how many boards are filled with formulas. The underlying problem and misunderstanding is here, that you can't put a value to a diverging sum. So putting a value to a sum does not mean it doesn't diverge. It means you spoiled the proof by swapping the prerequisite and the result of the proof. Submarines can swim, but that doesn't mean that swimmers are a submarine

[@John-eq8cu]

Not 'actually' -1/12 at all. You simply CANNOT use those tricks on a divergent series, so this result is total nonsense. But I know you already know that, so why are you still spending time on this?

[@trollkachu3744]

5:18 "So I will take you" :D

[@mp7311]

Saying 1+2+3+...=-1/12 is wrong unless specified to be using RS because unless you specify you are using a different definition, then it diverges.

[@misterlau5246]

With all due respect, don't take this as an attempt to argue or troll you. Ahem.. Cheater, lol! Mixing spaces, aren't you? N VS R. What is that, i? Complex issue it seems =) 8) Asigning values to an infinite divergent series. Ok. It is like an audio compressor. It reduces the intensity of the input signal above a threshold, and you have a knob to adjust compression ratio. 2:1, 8:1, infinite:1. But in practice, there's no infinite to 1 ratio. It just needs to be a big enough value, and that's the device's " infinite ". Strictly speaking, a series like 1+2+3 etc is not equal to a limited integral which you use. Cheerios!

[@GH-oi2jf]

It doesn't matter if Rumanujan thought of it, it's still nonsense. Either that, or you are misrepresenting his work.

[@chandankar5032]

Its a request to make video on euler mclaurin summation. No one has made it in KZbin yet. Please🙏

[@mayaq8324]

Problem with summation to infinity from a starting point, 1, is that infinity never ends, but never starts either, so the real take should be summation of All integers from minus infinity to plus infinity, and that results obviously in ZERO

[@RadixSort3]

what's deal with the round microphone. Please be kind to userself and get small attachable microphone. Even apple airpod should do.

[@batmanthedarkestknight]

"I didn't make this up " made me laugh so much

[@MS-cj8uw]

It's very easy to prove that this result is wrong. Just don't make shifting for any series that goes to infinity . because it's unlogic,

[@AJ-fo3hp]

Why it is giving different result with different approach But maths shouldn't be, different approach same result is valid proof Otherwise it is ambiguous

[@inyobill]

It doesn't make sense in the first, or the last place. It's at least equally valid that what the Numberphile mathematician demonstrated in the -1/12 video was that their assumption that: 1 -1 +1 -1 +1 -1 … = 1/2 is false. Assign values to divergent series only at risk.

[@thingthingthingthingthingthing]

Guys remember Riemann zeta function? -1 which is 1+2+3… seems to be going to -1/12

[@Crustyislooking]

JJDJEBCJSI I still hate this damn “sum”

[@blackpenredpen]

: )

[@ezxd5192]

And how do you know that Ramanujan is correct?

[@sxnchou]

he abso-fucking-lutely isnt. his theory is a load of bullshit because after learning fractions and negative numbers at third grade, and thought that those were the most advanced math in existence he decided to answer all his homework questions with a arbitrary negative fraction he pulled out of his ass. that fraction is -1/12

[@kabirahmed6984]

In your another video,you proved this sum is -1/8. But here is -1/12 with Ramanujan. What's the cruelty of the Infinity??? 🙄

[@SwvDo-e7e]

This is only possible if you assume that properties still work when you are comparing never ending sums right ?