1 + 2 + 3 + 4 + ⋯

Divergent series thumb|upright=1.35|The first four partial sums of the series 1 + 2 + 3 + 4 + ⋯. The parabola is their smoothed asymptote; its y-intercept is −1/12. |alt=A graph depicting the series with layered boxes and a parabola that dips just below the y-axis The infinite series whose terms are the positive integers 1 + 2 + 3 + 4 + ⋯ is a divergent series. The nth partial sum of the series is the triangular number which increases without bound as n goes to infinity. Because the sequence of partial sums fails to converge to a finite limit, the series does not have a sum. While the series itself diverges to infinity, in certain mathematical contexts it can be assigned a finite value. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of − 1 12, which is expressed by the famous formulaLepowsky J. 1999 Naihuan Jing and Kailash C. Misra Vertex operator algebras and the zeta function Recent Developments in Quantum Affine Algebras and Related Topics Contemporary Mathematics 248 327–340 math/9909178 1999math......9178L . where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned methods and not as the sum of an infinite series in its usual meaning. These methods have applications in other fields such as complex analysis, quantum field theory, and string theory.Tong David February 23, 2012 String Theory 0908.0333 28–48 hep-th In a monograph on moonshine theory, University of Alberta mathematician Terry Gannon calls this equation "one of the most remarkable formulae in science".Gannon Terry April 2010 Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics Cambridge University Press 978-0521141888 140. Partial sums Triangular number thumb|The first six triangular numbers The partial sums of the series 1 + 2 + 3 + 4 + 5 + 6 + ⋯ are 1, 3, 6, 10, 15, etc. The nth partial sum is given by a simple formula: This equation was known to the Pythagoreans as early as the sixth century BCE.Pengelley David J. 2002 The bridge between the continuous and the discrete via original sources Otto Bekken Study the Masters: The Abel-Fauvel Conference National Center for Mathematics Education, University of Gothenburg, Sweden 978-9185143009 3 etal Numbers of this form are called triangular numbers, because they can be arranged as an equilateral triangle. The infinite sequence of triangular numbers diverges to +∞, so by definition, the infinite series 1 + 2 + 3 + 4 + ⋯ also diverges to +∞. The divergence is a simple consequence of the form of the series: the terms do not approach zero, so the series diverges by the term test. Summability Among the classical divergent series, 1 + 2 + 3 + 4 + ⋯ is relatively difficult to manipulate into a finite value. Many summation methods are used to assign numerical values to divergent series, some more powerful than others. For example, Cesàro summation is a well-known method that sums Grandi's series, the mildly divergent series 1 − 1 + 1 − 1 + ⋯, to 2. Abel summation is a more powerful method that not only sums Grandi's series to 1 2, but also sums the trickier series 1 − 2 + 3 − 4 + ⋯ to 1 4. Unlike the above series, 1 + 2 + 3 + 4 + ⋯ is not Cesàro summable nor Abel summable. Those methods work on oscillating divergent series, but they cannot produce a finite answer for a series that diverges to +∞.Hardy 1949 10 Most of the more elementary definitions of the sum of a divergent series are stable and linear, and any method that is both stable and linear cannot sum 1 + 2 + 3 + ⋯ to a finite value (see Heuristics below). More advanced methods are required, such as zeta function regularization or Ramanujan summation. It is also possible to argue for the value of − 1 12 using some rough heuristics related to these methods. Heuristics thumb|upright=1.8|Passage from Ramanujan's first notebook describing the "constant" of the series Srinivasa Ramanujan presented two derivations of "1 + 2 + 3 + 4 + ⋯ = − 1 12" in chapter 8 of his first notebook.Ramanujan's Notebooks http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook1/chapterVIII/page3.htm January 26, 2014Wazir Hasan Abdi 1992 Toils and triumphs of Srinivasa Ramanujan, the man and the mathematician National 41Bruce C. Berndt 1985 Ramanujan's Notebooks: Part 1 Springer 135–136 The simpler argument proceeds in two steps, as follows. The first key insight is that the series of positive numbers 1 + 2 + 3 + 4 + ⋯ closely resembles the alternating series 1 − 2 + 3 − 4 + ⋯. The latter series is also divergent, but it is much easier to work with; there are several classical methods that assign it a value, which have been explored since the 18th century.Euler Leonhard Lucas Willis Thomas J. Osler Translation with notes of Euler's paper: Remarks on a beautiful relation between direct as well as reciprocal power series 2006 The Euler Archive http://www.math.dartmouth.edu/~euler/pages/E352.html 2007-03-22 Originally published as Euler Leonhard Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques fr Mémoires de l'Académie des Sciences de Berlin 1768 17 83–106 In order to transform the series 1 + 2 + 3 + 4 + ⋯ into 1 − 2 + 3 − 4 + ⋯, one can subtract 4 from the second term, 8 from the fourth term, 12 from the sixth term, and so on. The total amount to be subtracted is 4 + 8 + 12 + 16 + ⋯, which is 4 times the original series. These relationships can be expressed using algebra. Whatever the "sum" of the series might be, call it c = 1 + 2 + 3 + 4 + ⋯. Then multiply this equation by 4 and subtract the second equation from the first: The second key insight is that the alternating series 1 − 2 + 3 − 4 + ⋯ is the formal power series expansion (for x at point 0) of the function 1 (1 + x)2 which is 1 − 2x + 3x2 − 4x3 + ⋯ evaluated with x defined as 1. Accordingly, Ramanujan writes Dividing both sides by −3, one gets c = − 1 12. Generally speaking, it is incorrect to manipulate infinite series as if they were finite sums. For example, if zeroes are inserted into arbitrary positions of a divergent series, it is possible to arrive at results that are not self-consistent, let alone consistent with other methods. In particular, the step 4c = 0 + 4 + 0 + 8 + ⋯ is not justified by the additive identity law alone. For an extreme example, prepending a single zero to the front of the series can lead to a different result. One way to remedy this situation, and to constrain the places where zeroes may be inserted, is to keep track of each term in the series by attaching a dependence on some function.Promoting numbers to functions is identified as one of two broad classes of summation methods, including Abel and Borel summation, by Knopp Konrad Konrad Knopp Theory and Application of Infinite Series https://archive.org/details/theoryapplicatio00knop_448 limited 1990 1922 Dover 0-486-66165-2 475–476 In the series 1 + 2 + 3 + 4 + ⋯, each term n is just a number. If the term n is promoted to a function n−s, where s is a complex variable, then one can ensure that only like terms are added. The resulting series may be manipulated in a more rigorous fashion, and the variable s can be set to −1 later. The implementation of this strategy is called zeta function regularization. Zeta function regularization thumb|upright=1.4|Plot of ζ(s). For s > 1, the series converges and ζ(s) > 1. Analytic continuation around the pole at s = 1 leads to a region of negative values, including ζ(−1) = − 1 12. In zeta function regularization, the series is replaced by the series The latter series is an example of a Dirichlet series. When the real part of s is greater than 1, the Dirichlet series converges, and its sum is the Riemann zeta function ζ(s). On the other hand, the Dirichlet series diverges when the real part of s is less than or equal to 1, so, in particular, the series 1 + 2 + 3 + 4 + ⋯ that results from setting s = −1 does not converge. The benefit of introducing the Riemann zeta function is that it can be defined for other values of s by analytic continuation. One can then define the zeta-regularized sum of 1 + 2 + 3 + 4 + ⋯ to be ζ(−1). From this point, there are a few ways to prove that ζ(−1) = − 1 12. One method, along the lines of Euler's reasoning,Jeffrey Stopple 2003 A Primer of Analytic Number Theory: From Pythagoras to Riemann 0-521-81309-3 202 Cambridge University Press . uses the relationship between the Riemann zeta function and the Dirichlet eta function η(s). The eta function is defined by an alternating Dirichlet series, so this method parallels the earlier heuristics. Where both Dirichlet series converge, one has the identities: The identity continues to hold when both functions are extended by analytic continuation to include values of s for which the above series diverge. Substituting s −1, one gets −3ζ(−1) = η(−1). Now, computing η(−1) is an easier task, as the eta function is equal to the Abel sum of its defining series,Knopp Konrad Konrad Knopp Theory and Application of Infinite Series https://archive.org/details/theoryapplicatio00knop_448 limited 1990 1922 Dover 0-486-66165-2 490–492 which is a one-sided limit: Dividing both sides by −3, one gets ζ(−1) = − 1 12. Cutoff regularization Sum1234Plain.svg A graph depicting the series with layered boxes The series 1 + 2 + 3 + 4 + ⋯ Sum1234Smoothed.svg A graph depicting the smoothed series with layered curving stripes After smoothing thumb|Asymptotic behavior of the smoothing. The y-intercept of the parabola is − 1 12.Terence Tao Terence Tao April 10, 2010 The Euler–Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ January 30, 2014. |alt=A graph showing a parabola that dips just below the y-axis The method of regularization using a cutoff function can "smooth" the series to arrive at − 1 12. Smoothing is a conceptual bridge between zeta function regularization, with its reliance on complex analysis, and Ramanujan summation, with its shortcut to the Euler–Maclaurin formula. Instead, the method operates directly on conservative transformations of the series, using methods from real analysis. The idea is to replace the ill-behaved discrete series with a smoothed version where f is a cutoff function with appropriate properties. The cutoff function must be normalized to f(0) = 1; this is a different normalization from the one used in differential equations. The cutoff function should have enough bounded derivatives to smooth out the wrinkles in the series, and it should decay to 0 faster than the series grows. For convenience, one may require that f is smooth, bounded, and compactly supported. One can then prove that this smoothed sum is asymptotic to − 1 12 + CN2, where C is a constant that depends on f. The constant term of the asymptotic expansion does not depend on f: it is necessarily the same value given by analytic continuation, − 1 12. Ramanujan summation The Ramanujan sum of 1 + 2 + 3 + 4 + ⋯ is also − 1 12. Ramanujan wrote in his second letter to G. H. Hardy, dated 27 February 1913: Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. ... I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + ⋯ − 1 12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter.https://books.google.com/books?id=Of5G0r6DQiEC&pg=PA53 53 Ramanujan: Letters and Commentary 9780821891254 Aiyangar Srinivasa Ramanujan 7 September 1995 American Mathematical Soc. Ramanujan summation is a method to isolate the constant term in the Euler–Maclaurin formula for the partial sums of a series. For a function f, the classical Ramanujan sum of the series is defined as where f(2k−1) is the (2k − 1)th derivative of f and B2k is the (2k)th Bernoulli number: B2 = 1 6, B4 = − 1 30, and so on. Setting f(x) = x, the first derivative of f is 1, and every other term vanishes, soBruce C. Berndt 1985 Ramanujan's Notebooks: Part 1 Springer-Verlag 13, 134. To avoid inconsistencies, the modern theory of Ramanujan summation requires that f is "regular" in the sense that the higher-order derivatives of f decay quickly enough for the remainder terms in the Euler–Maclaurin formula to tend to 0. Ramanujan tacitly assumed this property. The regularity requirement prevents the use of Ramanujan summation upon spaced-out series like 0 + 2 + 0 + 4 + ⋯, because no regular function takes those values. Instead, such a series must be interpreted by zeta function regularization. For this reason, Hardy recommends "great caution" when applying the Ramanujan sums of known series to find the sums of related series.Hardy 1949 346 Failure of stable linear summation methods 1 + 1 + 1 + 1 + ⋯ A summation method that is linear and stable cannot sum the series 1 + 2 + 3 + ⋯ to any finite value. (Stable means that adding a term at the beginning of the series increases the sum by the value of the added term.) This can be seen as follows. If then adding 0 to both sides gives by stability. By linearity, one may subtract the second equation from the first (subtracting each component of the second line from the first line in columns) to give Adding 0 to both sides again gives and subtracting the last two series gives contradicting stability. Therefore, every method that gives a finite value to the sum 1 + 2 + 3 + ⋯ is not stable or not linear.Natiello Mario A. Solari Hernan Gustavo On the removal of infinities from divergent series Philosophy of Mathematics Education Journal 29 July 2015 1–11 11336/46148. Physics In bosonic string theory, the attempt is to compute the possible energy levels of a string, in particular, the lowest energy level. Speaking informally, each harmonic of the string can be viewed as a collection of D − 2 independent quantum harmonic oscillators, one for each transverse direction, where D is the dimension of spacetime. If the fundamental oscillation frequency is ω, then the energy in an oscillator contributing to the nth harmonic is nħω/2. So using the divergent series, the sum over all harmonics is −ħω(D − 2)/24. Ultimately it is this fact, combined with the Goddard–Thorn theorem, which leads to bosonic string theory failing to be consistent in dimensions other than 26.Bernardo Barbiellini 1987 The Casimir effect in conformal field theories Physics Letters B 190 1–2 137–139 10.1016/0370-2693(87)90854-9 1987PhLB..190..137B . The regularization of 1 + 2 + 3 + 4 + ⋯ is also involved in computing the Casimir force for a scalar field in one dimension.See v:Quantum mechanics/Casimir effect in one dimension.May 2021 An exponential cutoff function suffices to smooth the series, representing the fact that arbitrarily high-energy modes are not blocked by the conducting plates. The spatial symmetry of the problem is responsible for canceling the quadratic term of the expansion. All that is left is the constant term −1/12, and the negative sign of this result reflects the fact that the Casimir force is attractive.Zee 2003 pp. 65–67 A similar calculation is involved in three dimensions, using the Epstein zeta-function in place of the Riemann zeta function.Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists Eberhard Zeidler Springer 2007 9783540347644 305–306 https://books.google.com/books?id=XYtnGl9enNgC&pg=PA305 2006qftb.book.....Z. History It is unclear whether Leonhard Euler summed the series to − 1 12. According to Morris Kline, Euler's early work on divergent series relied on function expansions, from which he concluded 1 + 2 + 3 + 4 + ⋯ = ∞.Morris Kline Morris Kline November 1983 Euler and Infinite Series Mathematics Magazine 56 5 307–314 10.2307/2690371 2690371 . According to Raymond Ayoub, the fact that the divergent zeta series is not Abel-summable prevented Euler from using the zeta function as freely as the eta function, and he "could not have attached a meaning" to the series.Raymond Ayoub December 1974 Euler and the Zeta Function The American Mathematical Monthly 81 10 1067–1086 http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/RaymondAyoub.pdf February 14, 2014 10.2307/2319041 2319041 . Other authors have credited Euler with the sum, suggesting that Euler would have extended the relationship between the zeta and eta functions to negative integers.Lefort Jean Les séries divergentes chez Euler fr L'Ouvert IREM de Strasbourg 31 15–25 https://web.archive.org/web/20140222164751/http://irem.u-strasbg.fr/php/articles/31_Lefort_divergent.pdf February 22, 2014 http://irem.u-strasbg.fr/php/articles/31_Lefort_divergent.pdf February 14, 2014 dead Masanobu Kaneko Nobushige Kurokawa Masato Wakayama 2003 A variation of Euler's approach to values of the Riemann zeta function Kyushu Journal of Mathematics 57 1 175–192 10.2206/kyushujm.57.175 math/0206171 54514141 http://catalog.lib.kyushu-u.ac.jp/opac/repository/100000/handle/2324/11683/57_175.pdf January 31, 2014 https://web.archive.org/web/20140202114604/http://catalog.lib.kyushu-u.ac.jp/opac/repository/100000/handle/2324/11683/57_175.pdf 2014-02-02 dead Jonathan Sondow February 1994 Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series Proceedings of the American Mathematical Society 120 4 421–424 10.1090/S0002-9939-1994-1172954-7 https://www.ams.org/journals/proc/1994-120-02/S0002-9939-1994-1172954-7/ February 14, 2014 free In the primary literature, the series 1 + 2 + 3 + 4 + ⋯ is mentioned in Euler's 1760 publication la De seriebus divergentibus alongside the divergent geometric series 1 + 2 + 4 + 8 + ⋯. Euler hints that series of this type have finite, negative sums, and he explains what this means for geometric series, but he does not return to discuss 1 + 2 + 3 + 4 + ⋯. In the same publication, Euler writes that the sum of 1 + 1 + 1 + 1 + ⋯ is infinite.E. J. Barbeau P. J. Leah May 1976 Euler's 1760 paper on divergent series Historia Mathematica 3 2 141–160 10.1016/0315-0860(76)90030-6 In popular media David Leavitt's 2007 novel The Indian Clerk includes a scene where Hardy and Littlewood discuss the meaning of this series. They conclude that Ramanujan has rediscovered ζ(−1), and they take the "lunatic asylum" line in his second letter as a sign that Ramanujan is toying with them.David Leavitt David Leavitt 2007 The Indian Clerk Bloomsbury 61–62. Simon McBurney's 2007 play A Disappearing Number focuses on the series in the opening scene. The main character, Ruth, walks into a lecture hall and introduces the idea of a divergent series before proclaiming, "I'm going to show you something really thrilling", namely 1 + 2 + 3 + 4 + ⋯ = − 1 12. As Ruth launches into a derivation of the functional equation of the zeta function, another actor addresses the audience, admitting that they are actors: "But the mathematics is real. It's terrifying, but it's real."Simon McBurney April 2012 A Disappearing Number Oberon https://books.google.com/books?id=SYkHEAAAQBAJ 9781849432993 .Rachel Thomas December 1, 2008 A disappearing number Plus http://plus.maths.org/content/disappearing-number February 5, 2014 In January 2014, Numberphile produced a YouTube video on the series, which gathered over 1.5 million views in its first month.Dennis Overbye Dennis Overbye In the End, It All Adds Up to –1/12 https://www.nytimes.com/2014/02/04/science/in-the-end-it-all-adds-up-to.html February 3, 2014 The New York Times February 3, 2014 . The 8-minute video is narrated by Tony Padilla, a physicist at the University of Nottingham. Padilla begins with 1 − 1 + 1 − 1 + ⋯ and 1 − 2 + 3 − 4 + ⋯ and relates the latter to 1 + 2 + 3 + 4 + ⋯ using a term-by-term subtraction similar to Ramanujan's argument.w-I6XTVZXww ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ⋯ = –1/12. Numberphile also released a 21-minute version of the video featuring Nottingham physicist Ed Copeland, who describes in more detail how 1 − 2 + 3 − 4 + ⋯ = 1 4 as an Abel sum, and 1 + 2 + 3 + 4 + ⋯ = − 1 12 as ζ(−1).E-d9mgo8FGk Sum of Natural Numbers (second proof and extra footage). After receiving complaints about the lack of rigour in the first video, Padilla also wrote an explanation on his webpage relating the manipulations in the video to identities between the analytic continuations of the relevant Dirichlet series.Tony Padilla What do we get if we sum all the natural numbers? http://www.nottingham.ac.uk/~ppzap4/response.html February 3, 2014 September 12, 2021 https://web.archive.org/web/20210912051044/https://www.nottingham.ac.uk/~ppzap4/response.html dead In The New York Times coverage of the Numberphile video, mathematician Edward Frenkel commented: "This calculation is one of the best-kept secrets in math. No one on the outside knows about it." Coverage of this topic in Smithsonian magazine describes the Numberphile video as misleading and notes that the interpretation of the sum as − 1 12 relies on a specialized meaning for the equals sign, from the techniques of analytic continuation, in which equals means is associated with.http://www.smithsonianmag.com/smart-news/great-debate-over-whether-1234-112-180949559/ The Great Debate Over Whether 1 + 2 + 3 + 4... + ∞ = −1/12 Schultz Colin 2014-01-31 Smithsonian 2016-05-16 The Numberphile video was critiqued on similar grounds by German mathematician Burkard Polster on his Mathologer YouTube channel in 2018, his video receiving 2.7 million views by 2023.Polster Burkard January 13, 2018 Numberphile v. Math: the truth about 1+2+3+⋯=−1/12 https://www.youtube.com/watch?v=YuIIjLr6vUA August 31, 2023 YouTube References 30em Bibliography Berndt Bruce C. Srinivasa Ramanujan Aiyangar Srinivasa Ramanujan Robert A. Rankin Ramanujan: letters and commentary 1995 American Mathematical Society 0-8218-0287-9 Hardy G. H. G. H. Hardy Divergent Series 1949 Clarendon Press Zee A. Quantum field theory in a nutshell 2003 Princeton UP 0-691-01019-6 Further reading Zwiebach Barton A First Course in String Theory 2004 Cambridge UP 0-521-83143-1 See p. 293. Emilio Elizalde Cosmology: Techniques and Applications Proceedings of the II International Conference on Fundamental Interactions 2004 gr-qc/0409076 2004gr.qc.....9076E G. N. Watson April 1929 Theorems stated by Ramanujan (VIII): Theorems on Divergent Series Journal of the London Mathematical Society 1 4 2 82–86 10.1112/jlms/s1-4.14.82 External links divergent series Lamb E. (2014), "Does 1+2+3... Really Equal –1/12?", Scientific American Blogs. This Week's Finds in Mathematical Physics (Week 124), (Week 126), (Week 147), (Week 213) Euler's Proof That 1 + 2 + 3 + ⋯ = −1/12 – by John Baez http://math.ucr.edu/home/baez/numbers/24.pdf My Favorite Numbers: 24 John Baez September 19, 2008 John Baez The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation by Terence Tao A recursive evaluation of zeta of negative integers by Luboš Motl Link to Numberphile video 1 + 2 + 3 + 4 + 5 + ⋯ = –1/12 Sum of Natural Numbers (second proof and extra footage) includes demonstration of Euler's method. What do we get if we sum all the natural numbers? https://web.archive.org/web/20210912051044/https://www.nottingham.ac.uk/~ppzap4/response.html 2021-09-12 response to comments about video by Tony Padilla Related article from New York Times Why –1/12 is a gold nugget follow-up Numberphile video with Edward Frenkel Divergent Series: why 1 + 2 + 3 + ⋯ = −1/12 by Brydon Cais from University of Arizona Category:Divergent series Category:Arithmetic series