REAL TRUTH Behind Ramanujan Paradox | 1+2+3+4+....... = -1/12 | NMS Sir 

Ramanujan greatest mathematician the .Aur aise discussion per wo khush hote....kyonki ye pragati ki nisani hai.

Please don't spread what you don't know. When mathematicians say that the sum of all positive integers is \(-1/12\), they are referring to this value in the context of analytic continuation and Ramanujan summation, not in the traditional arithmetic sense. It’s a result that has profound implications in theoretical physics, particularly in string theory and quantum mechanics, where it helps to regularize otherwise divergent series. It is a profound result that bridges the gap between pure mathematics and theoretical physics, showing the power and applicability of advanced mathematical concepts in understanding the universe. In string theory, particularly in bosonic string theory, the concept of the sum \(-1/12\) appears naturally. When calculating the Casimir energy (the vacuum energy) of a string, this sum helps regularize infinities and yields meaningful physical results. **Casimir Effect:** In quantum field theory, the Casimir effect is a physical force arising from a quantized field. The \(-1/12\) value is essential in these calculations to ensure that the energy density between two conducting plates is finite and physically meaningful. In quantum field theory, regularization and renormalization techniques often lead to sums of divergent series. The \(-1/12\) result helps handle these divergences and provide finite results. **Regularization Techniques:** When calculating loop diagrams in QFT, divergent sums appear. Using techniques like zeta function regularization, the divergent sums can be assigned finite values, with \(-1/12\) being a classic example. Just for the sake of getting views, don't mislead students. Peace!!

This question actually has several interpretations. 1. This reference is related to definite integration. You may know the formula of ∑n = n(n+1)/2. If you plot the curve of this function as, y=(x^2+x)/2, you get a parabola, which is open from both left and right extreme. But there's a small bounded region in 3rd quadrant,whose area equals 1/12 units and as convention that it may be written as -1/12. 2. Ramanujan was a genius mathematician and he may have seen something that the current mathematical models were not enough to express those ideas. Also, because of lack of traditional educational, he developed different ways, signs of writing. So some mathematicians say that result is completely true, if you change the meaning of the symbol '='. Its not 'is equal to' its 'is associated with'. And then these equations hold true.

Sir by all respect you are wrong at 12:32 because every series whose last term is very small doesn't means that it will converge for example the harmonic sum decrease at every step still its divergent proved using integral test

Sir, Ramanujan was still not wrong. You are right that this is not possible in case of normal addition and standard calculus. But Ramanujan defined these sums in different mathematical context. Normal addition doesn't work for divergent series. Ramanujan invented a new way of summing, now called "Ramanujan summation" ( en.wikipedia.org/wiki/Ramanujan_summation ) to assign a value to divergent infinite series in a way that has interesting mathematical properties. In fact, this has some applications in quantum field theory and string theory for getting rid of infinities.

Ramanujan is not wrong in actual , because the summation he proposed is exactly true according to maths and is significant in higher dimensional Mathematics and physics ✔️ and many mathematicians later also found the same summation by different methods for example when you use ' Riemann Zetta function' to find sum of these sequence we actually get these results which Ramanujan Proposed and his summation is regarded as true as per Mathematics and also these summation are used in Quantum Physics . So although these sums are non intuitive but actually true when we deal with "Higher dimensional mathematics " 👍🏻✔️ .

Goat of maths---- ramanujan

Sir by respect you are wrong not totally wrong but in mathematics sir ramanujan's equation 1+2+3+4++...=-1/12 is right in rieman Zeta function rieman Zeta function use analytical method by rieman Zeta function sir ramanujan is right but by eular Zeta function sir ramanujan is wrong so this is paradox hope you read my comment sir😊

He was super genius.. Normal people even cant understand the way he did.. It is very far to find the logics behind his calculations..

Think about Riemann rearrangement theorem sir , he was not wrong. He had used rearrangement of oscilatory series. And if a series is divergent then it can be made to converge any real number

The idea was put forward by reimann using something called zeta function and analytical continuation . -1/12 is value of zeta function at input of -1 but zeta function has domain of (1,infinity) But if we use analytic continuation, we get -1/12

ये हमारा दुर्भाग्य है कि वे महापुरुष हमारे बीच नही हैं जो इस का उत्तर दे श्रीनिवास: रामनुजन

Hardy, G.H. Divergent Series. Oxford University Press.Hardy provides a detailed discussion on divergent series and the methods used to assign finite values to them, including Ramanujan's methods.Weinberg, S. The Quantum Theory of Fields. Cambridge University Press.This text discusses the use of zeta function regularization in quantum field theory.Tao, T. An Introduction to Measure Theory. American Mathematical Society.Tao’s notes and blog often provide accessible explanations for advanced topics, including the interpretation of divergent series in modern mathematics.Summary:The expression (1 + 2 + 3 + 4 + ... = -\frac{1}{12}) is a result of advanced techniques in summation and regularization, particularly through the Riemann zeta function. It’s not incorrect but rather part of a broader mathematical framework that extends beyond elementary series summation. In appropriate contexts, this result is both valid and useful, particularly in theoretical physics and advanced mathematics.

Heroism thrives in not just seclusion and acceptance of the theorem's given by geniuses but in thinking and creating n numbers of questions on those theorems... You're a great symbol of Hero who is is inspiring this generations to think and find our self intuition about th existance ... we respect you more than the then proff. Hardy who couldn't figure out these mistakes........ I'm a physics teacher andgot to say that yes i love u boss...and thank you

Sir now , I'm start releasing that you are doing all of this just because of views , You are saying only you are correct and rest don't know how to teach

Wonderful explanation sir ❤❤ so fortunate to have you ❤

Ramanujan was not the student of competitive exam preparation. These summation are in terms of analytic continuation not in usual arithmetic summation of natural numbers. Its widely used later by eminent theoretical physicist Richard Feynman and others to get rid of infinities and got Noble Prize later for his work in quantum electrodynamics, basically using the convergent integral at some point. Are you kidding that these people were not able to judge infinities. 😅😅😂😂😂 Its so hilarious that you are telling that the Mathematician Ramanujan didn't judge infinity correctly. He was not the coaching centre teacher who read and solved few previous year questions and books, and wrote these equations.

Series of natural number is closely linked to -1/12 but the conventional sum is not that connection

Its okay even newton said gravity is a force because it was best explaination . Then einstein said its the curvature of space

It is not usual sum which you are talking about first go and read advance mathematics and physics

Wonderful explanation Neelmani sir❤

Most beautiful explanation,i have ever seen❣️❣️❣️

thank thank you soooo much sir ji.........salo se ye mujhe presan kar rha tha🙏🙏🙏

Iska matlab string theory galat hai,jiska pahela e equation hai,Nms sir sahi hai,wah wah kya sir hai

Sir I am solving Resnick Halliday. Where to find Solutions???

Sir, it's not a little thing! because it's directly related to Riemann hypothesis through analytic continuation. ζ(s)=(2^s)(π^(s-1))(sin(πs/2))(Γ(1-s))(ζ(1-s)). if Riemann hypothesis will be proved true then, it will become 1+2+3+4+....... = -1/12. and there is one more thing- that's "1/(1-x)" series[where 0≤x

What an amazing myth buster... video ❤❤❤❤❤

Sir भले ही आप कोटा के teacher h पर ramanujan is real hero of math Support👍🏻👍🏻👍🏻

Very well sir that's great that teacher like you are still in you tube you are best teacher

Thank you sir.we need this kinds of discussions

Sir thanks 🙏🏻👍🏻 boht kuch sikhne ko mila sachme😊

Ramanujan sir hamare sath majak kar gaye 😅❤

Simple definition -infinity is something which is beyond human's imagination which has no limit.

Ramanujan was NOT wrong. He himself asked a question about this result. He himself asked if an extrapolation of summation of series could be done on divergent series(just like complex numbers for real numbers)? Basically, Ramanujan also proved it and threw it as a question to mathematicians, as if asking them to invent something new. He did NOT claim the result was true for the existing Mathematics.

Actually at 12:32, s = -1\9 is correct. In mathematics there is a field in which we can generalize real numbers as infinity called p-adic number or prime adic numbers for example: Taking base 10 (it's not a prime) We consider a number Say, ......11111111 = x X is 10-adic number Now we consider another number Y = .....99999 Now we add them 1 + Y = 1 + ...9999 = ......0000 Now as there are infinite zeros, we can never approach a 1 infront of it therefore we consider 1+Y ≈ 0 Therefore ....999 + 1 = 0 ...999 = -1 ....111111 = -1/9 X = - 1\9 ...1111 can be written as an infinite series ♾️... 1000 + 100 + 10 + 1 = -1/9 1+10+100+1000 ...... ♾️ = -1/9 So that prove in 12:33 is indeed a true statement, which is applicable in real numbers generalized as consisting infinite digits. Sir, plz tell if I am wrong.

Wah Maja aa Gaya Hats off to you Sir

Before discussing this kinds of things please read some higher level physics like Quantum mechanics , quantum field theory etc. There is an application of this series to understand THE CASIMIR FORCE.. dear students don't waste your time behind these.. To understand Ramanujan We need more advanced theory of physics.. But the problem of jee teacher is that their knowledge is restricted upto 12th Standard.. standard...

goat of physics-NMS SIR .I DONT STUDY FROM U BUT I LOVE TO WATCH YOU

Naya sikhne ko to yahi mila ki koi bhi kaise bhi kisi ko glt prove kr skta hai khud glt soch ke .ramanujan ne glti nahi ki hai vo maths ke gyata hai aur vo ye glti kaise krenge

When you add natural numbers of 10-adic number, it comes -1/12. But when you add natural numbers in prime-adic numbers, then this chaos doesn't occur and we get infinity as our answer. The problem is with the use of 10-adic numbers. Therefore, modern scientific community and computers use prime-adic numbers for scientific theory and calculations.

good everning sir🙏, sir ek question maine dekha tha jisme n^n^n... infinite tak ka value diya rheta tha(for example 26 ie n^n^n...=26) aur iske solution me diya tha ki n^26=26 aur through graph iska koi point exist krta tha. So is this valid tarika? Please help me Sir!!🤔🤔🥺

Well... The truth is this proof has been spread by youtube teachers. Ramanujan didn't actually give the value of this series. He instead gave the general formula for finding the Riemann Zeta Function for any value. Riemann Zeta Function is Z(x) = summation(1/n^x) where n goes from 1 to infinity. So, Z(1) becomes 1+1/2+1/3+... Z(2) becomes 1+1/2²+1/3²+... And so on. If you took calculus in college you would know a series of this form is convergent only for values >= 2. So, the domain of Z(x) itself is [2,infinity) But there were a few people who didn't seem to care much about the domain of functions so they started plugging in all sorts of values into the function given by Ramanujan and when they plugged in -1 they got -1/12. This is called analytic continuation in mathematics when you try to evaluate a function beyond it's domain AND THIS DOESN'T MEAN IT IS THE VALUE OF THE SERIES 1+2+3+.... A JEE student can understand this by relating to problems where you reject a solution for not satisfying the domain condition. For example let's say you were given an equation where you find x and you get the values 4 and -5. But the equation had a term log x in it so you dropped the -5 and only reported 4 because when you plug in -5 the log x term would go nuts. An algorithm only gives an output it's we who decide what values to keep and what to reject depending on the problem so plugging in the values in the original equation is crucial. This problem had a similar flaw. Ramanujan gave the formula to find Zeta function but people exploited the domain. Then there were some who created their own method to prove it is -1/12 so that general audience could understand it since they couldn't understand ramanujans formula as it involved many functions which are introduced after graduation. And that's where this proof came from. For a more detailed video on this and knowing the back story, you may checkout the video by BriTheMathGuy on the same. And yes the mistake in this solution is too trivial for someone like ramanujan so he could never come up with such a bogus approach. The first one to do so was numberphile.

Good job sir

Good explanation sir. Now all concepts are clear

Ramanuja a god gifted ❤🛐

Sir the sum -1/12 is found by analytical continuation of the function when you extend the input and output boundry of the function you get this value and yes this will not be used for normal day maths but for high level maths like in some high level therories of physics which discuss many diemension ie string theory etc

12:19 same proof was given in NCERT Class 10 in Mathematical Induction topic 7 years ago.

Salute ur great fundamental work

I was going to unsubscribe, i was really angry in start, because i am fan of sir Ramanujan, but i kept my angry, ego, aside and try to listen before doing anything, i know even sir Issac newton had made mistake, he had also said that light speed is more in denser medium and less in rarer medium, if newton can make mistake why not sir Ramanujan? I like your video, it take courage sir to say wrong to someone who is already master of that field, really sir you are doing great work. Thank you.

Current doesn’t follow vector addition laws, it follows KVL and KCL. Dot products of two vectors will be a scalar( I= J.A). The definition follows mathematics and no assumptions. How can you say that you call a quantity vector/scalar as per requirement?

without practicing of this segment you are here hats off you sir Error 10:34

It is not a mistake Sir, it is easy to flactuate students not teachers, Can I ask you one thing, are you a phd scholar of sequence and Series, khud achha padhao Sir ap bade teacher ho, Ramanujan k theory ko ulta sidha bol k kya siddh karne ki kosis kar rahe ho aap

Sir apko Mera pranam 🙏 Sir you are my role model❤

Yeh sir to har kisi mahan insan me bhi galti nikaal denge Inka to bs kaam hi yahi hai yeh bhaisaab apne aap lo Newton ka chacha samjhte hein

6÷2(1+2) कृपया इसका समाधान कीजिए गुरु जी।

Series S ek Gp thi with -1 as common ratio but for sum of infinite Gp r belongs to (-1;1) excluding both -1 and 1 so this wrong ❌

sir you are the one who explain it correctly I have same doubt that S and S +1 is not same untill it converging series which you have clear

Amazing explanation

Thanku sir

Mai to bio wla hun kuchh samjh nahi aaya but maja aaya😂

Nice explanation sir❤

Assumption :- S=infinity series Yani RHS term me hum koi bhi part me agar todte hai to wo part infinite series se chota hona chahiye.❤ S=x + y Jaha x aur y dono hi term S se chote honge

Fun fact: Ramanujan sir never proved it just stated that there may be a possibility that 1+2+3.. can also have value -1/12 other than infinity.

वैज्ञानिकों और गणितज्ञों के बाप ये हैं, वो लोग जीनियस नहीं थे सबसे बड़े जीनियस ये हैं😂

No no no 1-1+1-1+1.... It is a oscillatory series it's neither convergent nor divergent and 1+1+1+1+... It is a divergent series.Ramanujan's intuition was correct

I think ∞ serieses always shows some paradoxical nature. S is ∞ series. S = ½. Let S is a series with n terms. Then n ki koii aisi value nahi aayega ki S = ½ aaye. Hence its ∞ series.

इन्होंने कसम खायी है कि इस दुनिया मे सबको गलत ठहरा कर ही चैन लूंगा 😂😂😂😂 बिना सोचे समझे अपनी conditions लगा कर सबको गलत साबित कर देते है। अगला नंबर किसका है 😂😂😂😂 मुझे intazar है उस दिन का जब ये Eienstien को भी गलत साबित करेंगे 🤣🤣🤣🤣

Sir ek video on the topic "INFINITY" it always attracts me towards itself..... And your way of explaining complex things in a simple manner is amazing... Please if you can explain THE CONCEPT OF INFINITY... HOW SCIENTISTS WORKS ON IT.... WHAT WAS THE THOUGHT OF INFINITY SCIENTIST HAVE IN THERE MIND WHILE THEY WORK ON IT ?

When you consider infinity what you thing it it will be what ,on earth when you go further you sat you are going further but you will not say that you are going further. Like that .

Sir JEE advance ke baad ab bacho ko MTH101 padha rahe. IIT me kaam aayega ye

thanku sir , kal mai issi cheez ke baare me soch rha tha

First thank you sir 🙏 ki unka proof 1+2+3+....=-1/12 dikhaye , Infinite is undefined value how you let that last term is 1

sir , in Ramanujan series S = 1-1+1-1+1-1...... This series is not exploding to infinity . It just wobble around 0 and 1 ,so here S is not equal to infinity ,it is just any Real number .In this case we can cancel both sides of S as S is a finite number. However ,in yours series D = 1+1+1+1+1+..... which is obviously going to infinity ,so here we can not cancel out D (as infinity - infinity is not defined ) PLEASE EXPLAIN

Sir actually you know your 10000 jee advanced question came in ramanujam 1 question

Firsty sir you are not telling right this is not the whole intuition pf ramanujan firsty how is it become minus one by 1/12 firstly it is divergent series so it can't be converge ramanujan working on divergent series like 1/n is divergent series called harmonic series very fundamental example when we say series is convergent when it's partial sum sequence is convergent sum of natural number are like this divergent series because it's partial sum sequence is divergent but there is method called cesaro method to find limit of sequence when we apply this on partial sum sequence of natural number multiple time it answes minius 1/12 but this series is not actually convergent in normal sense but applying cesaros method it had to be converge forcefully so that we take a finite value the limit of partial sequence called sum of series by this it comes at this series is not useful in normal sense when ramanujan write this hardy firstly not understood this but it usages in mathematics come when putting complex number in reimamn jeta function this gives this series then we actually understood significance of this in complex analysis which is different thing apart from physics so yes it is not convergent series in normal sense but in special criteria it gives this answer which is very useful in mordern mathematics sor your explanation is not true ramanujan used ramanujan sum there are three kind of sum exist Abel sum cesaro sum and ramanujan sum I am trying to give brief idea of this I am doing msc in pure mathematics 🙏

Rule 1 that we know: Addition of positive numbers can't be negative, so 1+2+3+......... can't be any negative quantity

Sir please make video on all the contributions of ramanujan in various fields

Sir you don't mention several points which should be mentioned.😊

Infinity = continuous In series , not in parallel or branched Who can tell who has full knowledge and who knows what is happening in universe 0 means nothing Now 0/∞ is like big bang

Please respect Ramanujan Sir by saying Ramanujan sir and not only Ramanujan

Ramanujan sir be like:- it's prank guys Say hello to camera 📷

Reimann zeta function analytical conitunation is something that ramanujan talked about . and the proof is obviously wrong since the series is diverging and it is not given by ramanujan yet it is viral proof attached with the name of this great person.🙏

.Sir ek question hai infinity ka value hota hai kya 1/0 right or wrong 😊😊 aggar mai infinity ♾️ ka concept se dekho to nahi hota hai 😊kyu😊😊.

I get a lots of knowledge in the comment section than this person's video

Bhai mere 1-1+1-1+1..... Ka value 1/2 hota hai orr waha se sir ramanujan ke equation ko prove kiya ja sakta hai.

sir ramanujan uper se dekh rahe honge ki han duniya aage badh rahi hai

How you came with the conclusion in your example, 10S = 10+100+1000+.....♾️ Can you explain? How did you get the value of 10S?

Sir I laterally want the answer mene bhut dhunda pr kahi nhi mila

SIR DOUBT: app bol rahe hai infinity kahi land hoga lekin wo to chalta hi rehta hai wo kabhi rukta hi ni to app ne esa conclusion kyun kiya ki veh kahi end hoga aur infinity koi number to hai nahi ki hum bolde ye yaha end ho raha hai aur shayad 1+2+3+4...... = ND/ INFINITY ya 1-1+1-1+...... = S ab yeh (S) toh INFINITE time ke liye chalta hi jayega isme kabhi S = 0 kabhi S = 1

Agar general term ka domain extend kare toh ye thoda sense banata hai kyuki isme Bernoulli numbers ka bhi role hai aur reimann zeta function ka par us relationship ko define karna bahut kathin hoga

Sir what you are you as physics teachers and now math

Sir par ye maine suna hai ki use hota hai higher chizo me (ie. String theory) To kya ye duniya isse chalti hai😅😂

Sir, I have a suggestion for you. Have some more research on Analytic Continuation and zeros of Zeta function under complex analysis. Thank you... 🙌🏼

Answer is actually - 1/8 by correct series method

Any batches for droppers in online for JEE?

Sir ye sb toh thik hai lkin sir vo itne bade mathematician the unhe bhi toh ye cheez pta hogi hi naa ye dene se phele toh vo bhot si invention kr di thi na kya unhe nhi pta hoga ki ye toh impossible hai

Guru ke Guru hai aap❤❤

Sir, i heard his infinity sum [-1/12 ]is used in string theory

Sir it is not always true that if series last term is infinitely small then it is a conversing series .it may be conversing or diversing. As example harmoic series of nature number is a diversing series DIVERSING SERIES series that value is increasing when number of term is increased means if number of term is infinite then its value is also tending to infinity ♾️. CONVERSING SERIES series that have a finite value or when number of term is infinite then its tend to a finite value.

You don't have to explain it actually you got something wrong about your method without knowing actual way... How to prove it sir I love your physics teaching but don't mislead students providing them incomplete sense of maths..