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| [[File:Sum1234Summary.svg|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]]
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| The sum of all [[natural number]]s '''1 + 2 + 3 + 4 + · · ·''' is a [[divergent series]]. The ''n''th partial sum of the series is the [[triangular number]]
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| :<math>\sum_{k=1}^n k = \frac{n(n+1)}{2},</math>
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| which increases without bound as ''n'' goes to [[infinity]]. Because the [[sequence]] of partial sums fails to [[Limit of a sequence|converge to a finite limit]], the [[series (mathematics)|series]] is divergent, and it does not have a ''sum'' in the usual sense of the word.
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| Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of mathematically interesting results, some of which have applications in other fields such as [[complex analysis]], [[quantum field theory]] and [[string theory]]. There are many different [[summation method]]s used in mathematics to assign numerical values even to 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:<ref>{{Citation |last=Lepowsky |first=J. |date=1999 |editor=Naihuan Jing and Kailash C. Misra |title=Vertex operator algebras and the zeta function |booktitle=Recent Developments in Quantum Affine Algebras and Related Topics |series=Contemporary Mathematics |volume=248 |pages=327–340 |arxiv=math/9909178}}</ref>
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| :<math>1+2+3+4+\cdots=-\frac{1}{12}.</math>
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| In a monograph on [[moonshine theory]], Terry Gannon calls this equation "one of the most remarkable formulae in science".<ref>{{Citation |last=Gannon |first=Terry |date=April 2010 |title=Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics |publisher=Cambridge University Press |page=140}}</ref>
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| ==Partial sums==
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| [[File:First six triangular numbers.svg|thumb|The first six triangular numbers]]
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| {{Main|Triangular number}}
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| The partial sums of the series 1 + 2 + 3 + 4 + 5 + ⋯ are 1, 3, 6, 10, 15, etc. The ''n''th partial sum is given by a simple formula:
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| :<math>\sum_{k=1}^n k = \frac{n(n+1)}{2}</math>
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| This equation was known to the [[Pythagoreans]] as early as the sixth century B.C.E.<ref>{{Citation |last=Pengelley |first=David J. |date=2002 |title=The bridge between the continuous and the discrete via original sources |editor=Otto Bekken et al |booktitle=Study the Masters: The Abel-Fauvel Conference |publisher=National Center for Mathematics Education, University of Gothenburg, Sweden |page=3}}</ref> Numbers of this form are called [[triangular number]]s, because they can be arranged in a triangle.
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| 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]].
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| ==Summability==
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| Among the classical divergent series, {{nowrap|1 + 2 + 3 + 4 + · · ·}} is relatively difficult to manipulate into a finite value. There are many different [[summation method]]s 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 {{nowrap|1 − 1 + 1 − 1 + ⋯}}, to 1/2. [[Abel summation]] is a more powerful method that not only sums Grandi's series to 1/2, but also sums the trickier series {{nowrap|[[1 − 2 + 3 − 4 + · · ·]]}} to 1/4.
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| Unlike the above series, {{nowrap|1 + 2 + 3 + 4 + · · ·}} is not Cesàro summable or Abel summable. Those methods work only on convergent series and oscillating series; they cannot produce a finite answer for a series that diverges to +∞.<ref>Hardy p.10</ref> 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.
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| ===Heuristics===
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| [[Image:Ramanujan Notebook 1 Chapter 8 on 1234 series.jpg|thumb|upright=1.8|Passage from [[Srinivasa Ramanujan|Ramanujan]]'s first notebook describing the "constant" of the series]]
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| [[Srinivasa Ramanujan]] presented two derivations of "1 + 2 + 3 + 4 + ⋯ = −1/12" in chapter 8 of his first notebook.<ref>{{Citation |title=Ramanujan's Notebooks |url=http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook1/chapterVIII/page3.htm |accessdate=January 26, 2014}}</ref><ref>{{Citation |first=Wazir Hasan |last=Abdi |date=1992 |title=Toils and triumphs of Srinivasa Ramanujan, the man and the mathematician |publisher=National |page=41}}</ref><ref>{{Citation |first=Bruce C. |last=Berndt |date=1985 |title=Ramanujan’s Notebooks: Part 1 |publisher=Springer-Verlag |pages=135–136}}</ref> The simpler, less rigorous derivation proceeds in two steps, as follows.
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| The first key insight is that the series of positive numbers {{nowrap|1 + 2 + 3 + 4 + · · ·}} closely resembles the [[alternating series]] {{nowrap|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.<ref>{{cite web |author=Euler, Leonhard; Lucas Willis; and Thomas J Osler |title=Translation with notes of Euler's paper: Remarks on a beautiful relation between direct as well as reciprocal power series |year=2006 |publisher=The Euler Archive |url=http://www.math.dartmouth.edu/~euler/pages/E352.html |accessdate=2007-03-22}} Originally published as {{cite journal |last=Euler |first=Leonhard |title=Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques |journal=Memoires de l'academie des sciences de Berlin |year=1768 |volume=17 |pages=83–106}}</ref>
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| In order to transform the series {{nowrap|1 + 2 + 3 + 4 + · · ·}} into {{nowrap|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 {{nowrap|4 + 8 + 12 + 16 + · · ·}}, which is 4 times the original series. These relationships can be expressed with a bit of algebra. Whatever the "sum" of the series might be, call it {{nowrap|1=''c'' = 1 + 2 + 3 + 4 + ⋯.}} Then multiply this equation by 4 and subtract the second equation from the first:
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| :<math>
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| \begin{alignat}{7}
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| c&{}={}&1+2&&{}+3+4&&{}+5+6+\cdots \\
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| 4c&{}={}& 4&& {}+8&&{} +12+\cdots \\
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| -3c&{}={}&1-2&&{}+3-4&&{}+5-6+\cdots \\
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| \end{alignat}
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| </math>
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| The second key insight is that the alternating series {{nowrap|1 − 2 + 3 − 4 + · · ·}} is the formal power series expansion of the function 1/(1 + ''x'')<sup>2</sup> with 1 substituted for ''x''. Accordingly, Ramanujan writes:
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| :<math>-3c=1-2+3-4+\cdots=\frac{1}{(1+1)^2}=\frac14</math>
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| Dividing both sides by −3, one gets ''c'' = −1/12.
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| Generally speaking, it is dangerous to manipulate infinite series as if they were finite sums, and it is especially dangerous for divergent series. 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 {{nowrap|1=4''c'' = 0 + 4 + 0 + 8 + · · ·}} is not justified by the [[additive identity]] law alone. For an extreme example, appending a single zero to the front of the series can lead to inconsistent results.<ref name="Tao" />
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| 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.<ref>Promoting numbers to functions is identified as one of two broad classes of summation methods, including Abel and Borel summation, by {{cite book |last=Knopp |first=Konrad |authorlink=Konrad Knopp |title=Theory and Application of Infinite Series |year=1990 |origyear=1922 |publisher=Dover |isbn=0-486-66165-2 |pages=475–476}}</ref> In the series {{nowrap|1 + 2 + 3 + 4 + · · ·}}, each term ''n'' is just a number. If the term ''n'' is promoted to a function ''n''<sup>''−s''</sup>, 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]].
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| ===Zeta function regularization===
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| [[File:Zeta plot.gif|thumb|upright=1.4|Plot of ''ζ''(''s''). For {{nowrap|''s'' > 1}}, the series converges and {{nowrap|''ζ''(''s'') > 1}}. Analytic continuation around the pole at {{nowrap|1=''s'' = 1}} leads to a region of negative values, including {{nowrap|1=''ζ''(−1) = −1/12}}]]
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| In zeta function regularization, the series <math>\sum_{n=1}^\infty n</math> is replaced by the series <math>\sum_{n=1}^\infty n^{-s}</math>. 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 {{nowrap|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 {{nowrap|1 + 2 + 3 + 4 + · · ·}} to be ''ζ''(−1), which equals −1/12. (More generally, ''ζ''(''s'') will always be given by the degree zero term of the Laurent series expansion around ''h'' = 0 of <math>\sum_{n=1}^\infty {n^{-s}}e^{hn}</math>.)
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| There are a few different ways to prove that {{nowrap|1=''ζ''(−1) = −1/12.}} One method, along the lines of Euler's reasoning,<ref>{{Citation |first=Jeffrey |last=Stopple |date=2003 |title=A Primer of Analytic Number Theory: From Pythagoras to Riemann |isbn=0-521-81309-3 |page=202}}</ref> 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:
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| :<math>
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| \begin{alignat}{8}
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| \zeta(s)&{}={}&1^{-s}+2^{-s}&&{}+3^{-s}+4^{-s}&&{}+5^{-s}+6^{-s}+\cdots& \\
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| 2\cdot2^{-s}\zeta(s)&{}={}& 2\cdot2^{-s}&& {}+2\cdot4^{-s}&&{} +2\cdot6^{-s}+\cdots& \\
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| \left(1-2^{1-s}\right)\zeta(s)&{}={}&1^{-s}-2^{-s}&&{}+3^{-s}-4^{-s}&&{}+5^{-s}-6^{-s}+\cdots&=\eta(s) \\
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| \end{alignat}
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| </math>
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| The identity <math>(1-2^{1-s})\zeta(s)=\eta(s)</math> continues to hold when both functions are extended by analytic continuation to include values of ''s'' for which the above series diverge. Substituting {{nowrap|1=''s'' = −1}}, one gets {{nowrap|1=−3''ζ''(−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,<ref>{{cite book |last=Knopp |first=Konrad |authorlink=Konrad Knopp |title=Theory and Application of Infinite Series |year=1990 |origyear=1922 |publisher=Dover |isbn=0-486-66165-2 |pages=490–492}}
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| </ref> which is a [[one-sided limit]]:
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| :<math>-3\zeta(-1)=\eta(-1)=\lim_{x\nearrow 1}\left(1-2x+3x^2-4x^3+\cdots\right)=\lim_{x\nearrow 1}\frac{1}{(1+x)^2}=\frac14</math>
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| Dividing both sides by −3, one gets {{nowrap|1=''ζ''(−1) = −1/12.}}
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| ===Smoothed asymptotics===
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| {{Multiple image
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| |image1=Sum1234Plain.svg |alt1=A graph depicting the series with layered boxes |caption1=The series 1 + 2 + 3 + 4 + ⋯
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| |image2=Sum1234Smoothed.svg |alt2=A graph depicting the smoothed series with layered curving stripes |caption2=After smoothing
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| }}
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| [[File:Sum1234Asymptote.svg|thumb|Asymptotic behavior of the smoothing. The y-intercept of the parabola is −1/12.<ref name="Tao">{{Citation |first=Terence |last=Tao |authorlink=Terence Tao |date=April 10, 2010 |title=The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation |url=http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ |accessdate=January 30, 2014}}</ref> |alt=A graph showing a parabola that dips just below the y-axis]]
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| [[Terence Tao]] has described a method of smoothing 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]].
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| The idea is to replace the ill-behaved discrete series <math>\sum_{n=0}^Nn</math> with a smoothed version <math>\sum_{n=0}^\infty nf(n/N)</math>, where ''f'' is a [[cutoff function]] with appropriate properties. The cutoff function must be normalized to {{nowrap|1=''f''(0) = 1}}; this is a different normalization than 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 function|smooth]], [[bounded function|bounded]], and [[compactly supported]]. One can then prove that this smoothed sum is [[asymptotic]] to {{nowrap|−1/12 + ''CN''<sup>2</sup>}}, 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.<ref name="Tao" />
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| ===Ramanujan summation===
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| The [[Ramanujan summation|Ramanujan sum]] of {{nowrap|1 + 2 + 3 + 4 + · · ·}} is also −1/12. In Ramanujan's second letter to [[G. H. Hardy]], dated 27 February 1913, he wrote:
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| :"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 [[Thomas John I'Anson Bromwich|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: {{nowrap|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. …"<ref>Berndt et al. [http://books.google.com/books?id=Of5G0r6DQiEC&pg=PA53&dq=gratified p.53.]</ref>
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| ==Physics==
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| 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 <math>D-2</math> independent [[quantum harmonic oscillator]]s, one for each [[transverse wave|transverse direction]], where <math>D</math> is the dimension of spacetime. If the fundamental oscillation frequency is <math>\omega</math> then the energy in an oscillator contributing to the <math>n</math>th harmonic is <math>n\hbar\omega/2</math>. So using the divergent series, the sum over all harmonics is <math>-\hbar\omega (D-2)/24</math>. 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.
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| The regularization of 1 + 2 + 3 + 4 + ⋯ is also involved in computing the [[Casimir force#Derivation of Casimir effect assuming zeta-regularization|Casimir force]] for a [[scalar field]] in one dimension. 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.<ref>Zee pp.65–67</ref>
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| A similar calculation is involved in three dimensions, using the [[Real analytic Eisenstein series#Epstein zeta function|Epstein zeta-function]] in place of the Riemann zeta function.<ref>{{citation|title=Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists|first=Eberhard|last=Zeidler|publisher=Springer|year=2007|isbn=9783540347644|pages=305–306|url=http://books.google.com/books?id=XYtnGl9enNgC&pg=PA305}}.</ref>
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| ==In popular media==
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| [[David Leavitt]]'s 2007 novel ''[[The Indian Clerk]]'' includes a scene where Hardy and [[John Edensor Littlewood|Littlewood]] discuss the meaning of this series.<ref>{{Citation |first=David |last=Leavitt |authorlink=David Leavitt |date=2007 |title=The Indian Clerk |publisher=Bloomsbury |pages=61–62}}</ref>
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| In January 2014, the [[YouTube]] series [[Numberphile]] produced a video on the series, which gathered over 1.5 millions views in its first month.<ref name="Overbye">{{Citation |first=Dennis |last=Overbye |date=February 3, 2014 |title=In the End, It All Adds Up to – 1/12 |url=http://www.nytimes.com/2014/02/04/science/in-the-end-it-all-adds-up-to.html |accessdate=February 3, 2014 |work=New York TImes}}</ref> 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.<ref>{{YouTube |id=w-I6XTVZXww |title=ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12}}</ref> 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).<ref>{{YouTube |id=E-d9mgo8FGk |title=Sum of Natural Numbers (second proof and extra footage)}}</ref> After receiving complaints about the lack of rigor 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.<ref>{{Citation |first=Tony |last=Padilla |title=What do we get if we sum all the natural numbers? |url=http://www.nottingham.ac.uk/~ppzap4/response.html |accessdate=February 3, 2014}}</ref>
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| ==Notes==
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| {{reflist}}
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| ==References==
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| <div class="references-small">
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| *{{cite book |author=Berndt, Bruce C., Srinivasa Ramanujan Aiyangar, and Robert A. Rankin |title=Ramanujan: letters and commentary |year=1995 |publisher=American Mathematical Society |isbn=0-8218-0287-9}}
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| *{{cite book |last=Hardy |first=G.H. |authorlink=G. H. Hardy |title=Divergent Series |year=1949 |publisher=Clarendon Press |id={{LCC|QA295|.H29|1967}}}}
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| *{{cite book |last=Zee |first=A. |title=Quantum field theory in a nutshell |year=2003 |publisher=Princeton UP |isbn=0-691-01019-6}}
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| </div>
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| ==Further reading==
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| *{{cite journal |last=Lepowsky |first=James |title=Vertex operator algebras and the zeta function |journal=Contemporary Mathematics |volume=248 |year=1999 |pages=327–340 |arxiv=math/9909178}}
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| *{{cite book |last=Zwiebach |first=Barton |title=A First Course in String Theory |year=2004 |publisher=Cambridge UP |isbn=0-521-83143-1}} See p. 293.
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| *{{Citation |first=Masanobu |last=Kaneko |first2=Nobushige |last2=Kurokawa |first3=Masato |last3=Wakayama |date=2003 |title=A variation of Euler's approach to values of the Riemann zeta function |journal=Kyushu Journal of Mathematics |volume=57 |issue=1 |pages=175–192 |doi=10.2206/kyushujm.57.175 |arxiv=0206171 |url=http://catalog.lib.kyushu-u.ac.jp/opac/repository/100000/handle/2324/11683/57_175.pdf |accessdate=January 31, 2014}}
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| *{{cite conference |first=Emilio |last=Elizalde |title=Cosmology: Techniques and Applications |booktitle=Proceedings of the II International Conference on Fundamental Interactions |year=2004 |arxiv=gr-qc/0409076}}
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| *{{Citation |first=G. N. |last=Watson |date=April 1929 |title=Theorems stated by Ramanujan (VIII): Theorems on Divergent Series |journal=Journal of the London Mathematical Society |series=1 |volume=4 |issue=2 |pages=82–86 |doi=10.1112/jlms/s1-4.14.82}}
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| ==External links==
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| *[http://math.ucr.edu/home/baez/week124.html This Week's Finds in Mathematical Physics (Week 124)], [http://math.ucr.edu/home/baez/week126.html (Week 126)], [http://math.ucr.edu/home/baez/week147.html (Week 147)], [http://math.ucr.edu/home/baez/week213.html (Week 213)]
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| **[http://math.ucr.edu/home/baez/qg-winter2004/zeta.pdf Euler’s Proof That 1 + 2 + 3 + · · · = −1/12] - By [[John Baez]]
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| **{{cite web|url=http://math.ucr.edu/home/baez/numbers/24.pdf|title=My Favorite Numbers: 24|author=[[John Baez]]|date=September 19, 2008}}
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| *[http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation] by [[Terence Tao]]
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| * [http://motls.blogspot.co.uk/2014/01/a-recursive-evaluation-of-zeta-of.html A recursive evaluation of zeta of negative integers] by [[Luboš Motl]]
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| * [http://www.youtube.com/watch?v=w-I6XTVZXww ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12] [[Numberphile]] video with over a million views
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| ** [http://www.youtube.com/watch?v=E-d9mgo8FGk&feature=youtu.be Sum of Natural Numbers (second proof and extra footage)] includes demonstration of Euler's method.
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| ** [http://www.nottingham.ac.uk/~ppzap4/response.html What do we get if we sum all the natural numbers?] response to comments about video by [[Tony Padilla]]
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| ** [http://www.nytimes.com/2014/02/04/science/in-the-end-it-all-adds-up-to.html?hpw&rref=science Related article from New York TImes]
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| * [http://math.arizona.edu/~cais/Papers/Expos/div.pdf Divergent Series: why 1 + 2 + 3 + · · · = −1/12] by Brydon Cais from University of Arizona
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| {{Series (mathematics)}}
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| {{DEFAULTSORT:1 + 2 + 3 + 4 + ...}}
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| [[Category:Divergent series]]
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| [[Category:Arithmetic series]]
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