1+2+3+4+5+6

A graph depicting the series with layered boxes and a parabola that dips just below the y-axis
The first four partial sums of the series 1 + 2 + 3 + 4 + ⋯. The parabola is their smoothed asymptote; its y-intercept is −1/12.[1]

The infinite series whose terms are the natural numbers 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 lớn infinity. Because the sequence of partial sums fails to lớn converge to lớn a finite limit, the series does not have a sum.

Bạn đang xem: 1+2+3+4+5+6

Although the series seems at first sight not to lớn have any meaningful value at all, it can be manipulated to lớn yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to lớn assign numerical values even to lớn a divergent series. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of +1/12, which is expressed by a famous formula:[2]

where the left-hand side has to lớn be interpreted as being the value obtained by using one of the aforementioned summation 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.[3]

In a monograph on moonshine theory, University of Alberta mathematician Terry Gannon calls this equation "one of the most remarkable formulae in science".[4]

Partial sums[edit]

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 lớn the Pythagoreans as early as the sixth century BCE.[5] Numbers of this sườn are called triangular numbers, because they can be arranged as an equilateral triangle.

The infinite sequence of triangular numbers diverges to lớn +∞, sánh by definition, the infinite series 1 + 2 + 3 + 4 + ⋯ also diverges to lớn +∞. The divergence is a simple consequence of the sườn of the series: the terms tự not approach zero, sánh the series diverges by the term test.

Summability[edit]

Among the classical divergent series, 1 + 2 + 3 + 4 + ⋯ is relatively difficult to lớn manipulate into a finite value. Many summation methods are used to lớn assign numerical values to lớn divergent series, some more powerful phàn nàn others. For example, Cesàro summation is a well-known method that sums Grandi's series, the mildly divergent series 1 − 1 + 1 − 1 + ⋯, to lớn 1/2. Abel summation is a more powerful method that not only sums Grandi's series to lớn 1/2, but also sums the trickier series 1 − 2 + 3 − 4 + ⋯ to lớn 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 lớn +∞.[6] 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 lớn a finite value; see below. More advanced methods are required, such as zeta function regularization or Ramanujan summation. It is also possible to lớn argue for the value of +1/12 using some rough heuristics related to lớn these methods.

Heuristics[edit]

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.[7][8][9] The simpler, less rigorous derivation 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 lớn work with; there are several classical methods that assign it a value, which have been explored since the 18th century.[10]

In order to lớn 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 sánh on. The total amount to lớn 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, Gọi 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 of the function 1/(1 + x)2 but with x defined as 1. (This can be seen by equating 1/1 + x to lớn the alternating sum of the nonnegative powers of x, and then differentiating and negating both sides of the equation.) Accordingly, Ramanujan writes

Dividing both sides by −3, one gets c = +1/12.

Generally speaking, it is incorrect to lớn 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 lớn 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, appending a single zero to lớn the front of the series can lead to lớn a different result.[1]

One way to lớn remedy this situation, and to lớn constrain the places where zeroes may be inserted, is to lớn keep track of each term in the series by attaching a dependence on some function.[11] In the series 1 + 2 + 3 + 4 + ⋯, each term n is just a number. If the term n is promoted to lớn a function n−s, where s is a complex variable, then one can ensure that only lượt thích terms are added. The resulting series may be manipulated in a more rigorous fashion, and the variable s can be phối to lớn −1 later. The implementation of this strategy is called zeta function regularization.

Zeta function regularization[edit]

Plot of ζ(s). For s > 1, the series converges and ζ(s) > 1. Analytic continuation around the pole at s = 1 leads to lớn 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 phàn nàn 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 phàn nàn or equal to lớn 1, sánh, 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 lớn be ζ(−1).

From this point, there are a few ways to lớn prove that ζ(−1) = +1/12. One method, along the lines of Euler's reasoning,[12] 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, sánh this method parallels the earlier heuristics. Where both Dirichlet series converge, one has the identities:

The identity continues to lớn hold when both functions are extended by analytic continuation to lớn 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 lớn the Abel sum of its defining series,[13] which is a one-sided limit:

Dividing both sides by −3, one gets ζ(−1) = +1/12.

Cutoff regularization[edit]

A graph depicting the series with layered boxes

The series 1 + 2 + 3 + 4 + ⋯

A graph depicting the smoothed series with layered curving stripes

After smoothing

A graph showing a parabola that dips just below the y-axis
Asymptotic behavior of the smoothing. The y-intercept of the parabola is +1/12.[1]

The method of regularization using a cutoff function can "smooth" the series to lớn 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 lớn the Euler–Maclaurin formula. Instead, the method operates directly on conservative transformations of the series, using methods from real analysis.

The idea is to lớn replace the ill-behaved discrete series with a smoothed version

Xem thêm: tâm đường tròn nội tiếp

where f is a cutoff function with appropriate properties. The cutoff function must be normalized to lớn f(0) = 1; this is a different normalization from the one used in differential equations. The cutoff function should have enough bounded derivatives to lớn smooth out the wrinkles in the series, and it should decay to lớn 0 faster phàn nàn 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 lớn +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.[1]

Ramanujan summation[edit]

The Ramanujan sum of 1 + 2 + 3 + 4 + ⋯ is also +1/12. Ramanujan wrote in his second letter to lớn 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 lớn the one which a Mathematics Professor at London wrote asking má to lớn 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 lớn má the lunatic asylum as my goal. I dilate on this simply to lớn convince you that you will not be able to lớn follow my methods of proof if I indicate the lines on which I proceed in a single letter. ..."[14]

Ramanujan summation is a method to lớn 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 sánh on. Setting f(x) = x, the first derivative of f is 1, and every other term vanishes, so[15]

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 lớn tend to lớn 0. Ramanujan tacitly assumed this property.[15] The regularity requirement prevents the use of Ramanujan summation upon spaced-out series lượt thích 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 lớn find the sums of related series.[16]

Failure of stable linear summation methods[edit]

A summation method that is linear and stable cannot sum the series 1 + 2 + 3 + ⋯ to lớn 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 lớn 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 lớn give

Adding 0 to lớn both sides again gives

and subtracting the last two series gives

contradicting stability.

Therefore, every method that gives a finite value to lớn the sum 1 + 2 + 3 + ⋯ is not stable or not linear.[17]

Physics[edit]

In bosonic string theory, the attempt is to lớn 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 lớn the n-th 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 lớn bosonic string theory failing to lớn be consistent in dimensions other phàn nàn 26.[18]

The regularization of 1 + 2 + 3 + 4 + ⋯ is also involved in computing the Casimir force for a scalar field in one dimension.[19] An exponential cutoff function suffices to lớn 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.[20]

A similar calculation is involved in three dimensions, using the Epstein zeta-function in place of the Riemann zeta function.[21]

History[edit]

It is unclear whether Leonhard Euler summed the series to lớn +1/12. According to lớn Morris Kline, Euler's early work on divergent series relied on function expansions, from which he concluded 1 + 2 + 3 + 4 + ⋯ = ∞.[22] According to lớn 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 lớn the series.[23] Other authors have credited Euler with the sum, suggesting that Euler would have extended the relationship between the zeta and eta functions to lớn negative integers.[24][25][26] In the primary literature, the series 1 + 2 + 3 + 4 + ⋯ is mentioned in Euler's 1760 publication 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 lớn discuss 1 + 2 + 3 + 4 + ⋯. In the same publication, Euler writes that the sum of 1 + 1 + 1 + 1 + ⋯ is infinite.[27]

In popular media[edit]

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.[28]

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 lớn 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."[29][30]

Xem thêm: so sánh hướng động và ứng động

In January năm trước, Numberphile produced a YouTube video clip on the series, which gathered over 1.5 million views in its first month.[31] The 8-minute video clip 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 lớn 1 + 2 + 3 + 4 + ⋯ using a term-by-term subtraction similar to lớn Ramanujan's argument.[32] Numberphile also released a 21-minute version of the video clip 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).[33] After receiving complaints about the lack of rigour in the first video clip, Padilla also wrote an explanation on his webpage relating the manipulations in the video clip to lớn identities between the analytic continuations of the relevant Dirichlet series.[34]

In The Thành Phố New York Times coverage of the Numberphile video clip, mathematician Edward Frenkel commented: "This calculation is one of the best-kept secrets in math. No one on the outside knows about it."[31]

Coverage of this topic in Smithsonian magazine describes the Numberphile video clip 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.[35] The Numberphile video clip was critiqued on similar grounds by German mathematician Burkard Polster on his Mathologer YouTube channel in 2018, his video clip receiving 2.7 million views by 2023.[36]

References[edit]

  1. ^ a b c d Tao, Terence (April 10, 2010), The Euler–Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, retrieved January 30, 2014.
  2. ^ Lepowsky, J. (1999). "Vertex operator algebras and the zeta function". In Naihuan Jing and Kailash C. Misra (ed.). Recent Developments in Quantum Affine Algebras and Related Topics. Contemporary Mathematics. Vol. 248. pp. 327–340. arXiv:math/9909178. Bibcode:1999math......9178L..
  3. ^ Tong, David (February 23, 2012). "String Theory". pp. 28–48. arXiv:0908.0333 [hep-th].
  4. ^ Gannon, Terry (April 2010), Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, Cambridge University Press, p. 140, ISBN 978-0521141888.
  5. ^ Pengelley, David J. (2002). "The bridge between the continuous and the discrete via original sources". In Otto Bekken; et al. (eds.). Study the Masters: The Abel-Fauvel Conference. National Center for Mathematics Education, University of Gothenburg, Sweden. p. 3. ISBN 978-9185143009..
  6. ^ Hardy 1949, p. 10.
  7. ^ Ramanujan's Notebooks, retrieved January 26, 2014
  8. ^ Abdi, Wazir Hasan (1992), Toils and triumphs of Srinivasa Ramanujan, the man and the mathematician, National, p. 41
  9. ^ Berndt, Bruce C. (1985), Ramanujan's Notebooks: Part 1, Springer-Verlag, pp. 135–136
  10. ^ Euler, Leonhard (2006). "Translation with notes of Euler's paper: Remarks on a beautiful relation between direct as well as reciprocal power series". Translated by Willis, Lucas; Osler, Thomas J. The Euler Archive. Retrieved 2007-03-22. Originally published as Euler, Leonhard (1768). "Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques". Mémoires de l'Académie des Sciences de Berlin (in French). 17: 83–106.
  11. ^ Promoting numbers to lớn functions is identified as one of two broad classes of summation methods, including Abel and Borel summation, by Knopp, Konrad (1990) [1922]. Theory and Application of Infinite Series. Dover. pp. 475–476. ISBN 0-486-66165-2.
  12. ^ Stopple, Jeffrey (2003), A Primer of Analytic Number Theory: From Pythagoras to lớn Riemann, p. 202, ISBN 0-521-81309-3.
  13. ^ Knopp, Konrad (1990) [1922]. Theory and Application of Infinite Series. Dover. pp. 490–492. ISBN 0-486-66165-2.
  14. ^ Aiyangar, Srinivasa Ramanujan (7 September 1995). Ramanujan: Letters and Commentary. p. 53. ISBN 9780821891254.
  15. ^ a b Berndt, Bruce C. (1985), Ramanujan's Notebooks: Part 1, Springer-Verlag, pp. 13, 134.
  16. ^ Hardy 1949, p. 346.
  17. ^ Natiello, Mario A.; Solari, Hernan Gustavo (July 2015), "On the removal of infinities from divergent series", Philosophy of Mathematics Education Journal, 29: 1–11, hdl:11336/46148.
  18. ^ Barbiellini, Bernardo (1987), "The Casimir effect in conformal field theories", Physics Letters B, 190 (1–2): 137–139, Bibcode:1987PhLB..190..137B, doi:10.1016/0370-2693(87)90854-9.
  19. ^ See v:Quantum mechanics/Casimir effect in one dimension.[unreliable source?]
  20. ^ Zee 2003, pp. 65–67.
  21. ^ Zeidler, Eberhard (2007), "Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists", Quantum Field Theory I: Basics in Mathematics and Physics. A Bridge Between Mathematicians and Physicists, Springer: 305–306, Bibcode:2006qftb.book.....Z, ISBN 9783540347644.
  22. ^ Kline, Morris (November 1983), "Euler and Infinite Series", Mathematics Magazine, 56 (5): 307–314, CiteSeerX 10.1.1.639.6923, doi:10.2307/2690371, JSTOR 2690371.
  23. ^ Ayoub, Raymond (December 1974), "Euler and the Zeta Function" (PDF), The American Mathematical Monthly, 81 (10): 1067–1086, doi:10.2307/2319041, JSTOR 2319041, retrieved February 14, 2014.
  24. ^ Lefort, Jean, "Les séries divergentes chez Euler" (PDF), L'Ouvert (in French), IREM de Strasbourg (31): 15–25, archived from the original (PDF) on February 22, 2014, retrieved February 14, 2014.
  25. ^ Kaneko, Masanobu; Kurokawa, Nobushige; Wakayama, Masato (2003), "A variation of Euler's approach to lớn values of the Riemann zeta function" (PDF), Kyushu Journal of Mathematics, 57 (1): 175–192, arXiv:math/0206171, doi:10.2206/kyushujm.57.175, S2CID 54514141, archived from the original (PDF) on 2014-02-02, retrieved January 31, 2014.
  26. ^ Sondow, Jonathan (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, doi:10.1090/S0002-9939-1994-1172954-7, retrieved February 14, 2014.
  27. ^ Barbeau, E. J.; Leah, P.. J. (May 1976), "Euler's 1760 paper on divergent series", Historia Mathematica, 3 (2): 141–160, doi:10.1016/0315-0860(76)90030-6.
  28. ^ Leavitt, David (2007), The Indian Clerk, Bloomsbury, pp. 61–62.
  29. ^ Complicite (April 2012), A Disappearing Number, Oberon, ISBN 9781849432993.
  30. ^ Thomas, Rachel (December 1, 2008), "A disappearing number", Plus, retrieved February 5, 2014.
  31. ^ a b Overbye, Dennis (February 3, 2014), "In the End, It All Adds Up to lớn –1/12", The Thành Phố New York Times, retrieved February 3, 2014.
  32. ^ ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = –1/12 on YouTube.
  33. ^ Sum of Natural Numbers (second proof and extra footage) on YouTube.
  34. ^ Padilla, Tony, What tự we get if we sum all the natural numbers?, retrieved February 3, 2014.
  35. ^ Schultz, Colin (2014-01-31). "The Great Debate Over Whether 1 + 2 + 3 + 4... + ∞ = −1/12". Smithsonian. Retrieved 2016-05-16.
  36. ^ Polster, Burkard (January 13, 2018). Numberphile v. Math: the truth about 1+2+3+...=-1/12. Retrieved August 31, 2023 – via YouTube.

Bibliography[edit]

Further reading[edit]

  • Zwiebach, Barton (2004). A First Course in String Theory. Cambridge UP. ISBN 0-521-83143-1. See p. 293.
  • Elizalde, Emilio (2004). "Cosmology: Techniques and Applications". Proceedings of the II International Conference on Fundamental Interactions. arXiv:gr-qc/0409076. Bibcode:2004gr.qc.....9076E.
  • Watson, G. N. (April 1929), "Theorems stated by Ramanujan (VIII): Theorems on Divergent Series", Journal of the London Mathematical Society, 1, 4 (2): 82–86, doi:10.1112/jlms/s1-4.14.82

External links[edit]

  • 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
    • John Baez (September 19, 2008). "My Favorite Numbers: 24" (PDF).
  • 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 lớn Numberphile video clip 1 + 2 + 3 + 4 + 5 + ... = –1/12
    • Sum of Natural Numbers (second proof and extra footage) includes demonstration of Euler's method.
    • What tự we get if we sum all the natural numbers? response to lớn comments about video clip by Tony Padilla
    • Related article from Thành Phố New York Times
    • Why –1/12 is a gold nugget follow-up Numberphile video clip with Edward Frenkel
  • Divergent Series: why 1 + 2 + 3 + ⋯ = −1/12 by Brydon Cais from University of Arizona