1841:
20:
305:
2224:
5120:
2237:
1366:
2209:
5429:
3433:. 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. 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. In the primary literature, the series
1836:{\displaystyle {\begin{alignedat}{7}\zeta (s)&{}={}&1^{-s}+2^{-s}&&{}+3^{-s}+4^{-s}&&{}+5^{-s}+6^{-s}+\cdots &\\2\times 2^{-s}\zeta (s)&{}={}&2\times 2^{-s}&&{}+2\times 4^{-s}&&{}+2\times 6^{-s}+\cdots &\\\left(1-2^{1-s}\right)\zeta (s)&{}={}&1^{-s}-2^{-s}&&{}+3^{-s}-4^{-s}&&{}+5^{-s}-6^{-s}+\cdots &=\eta (s).\end{alignedat}}}
1106:
573:
855:
3390:
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
1053:
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,
522:
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 +∞. Most of the more elementary definitions of the sum of a divergent series are stable and linear, and any method that is both stable
653:
2165:
2430:; 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
2986:, 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.
2579:
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.
3595:(−1). 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.
2788:
850:{\displaystyle {\begin{alignedat}{7}c={}&&1+2&&{}+3+4&&{}+5+6+\cdots \\4c={}&&4&&{}+8&&{}+12+\cdots \\c-4c={}&&1-2&&{}+3-4&&{}+5-6+\cdots \end{alignedat}}}
2981:
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
1955:
3496:
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
3519:. 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."
1371:
1022:
658:
2968:
2549:"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
2414:
405:
141:
2642:
1069:
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. In the series
264:
3328:, 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
1916:
2346:
1268:
625:. 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.
4875:
3688:
3124:
3186:
1360:). 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:
1214:
3304:
3245:
3056:
169:
Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of different mathematical results. For example, many
3129:
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
3008:
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
2650:
4332:
1089:
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
5311:
4135:
3546:
using a term-by-term subtraction similar to
Ramanujan's argument. Numberphile also released a 21-minute version of the video featuring Nottingham physicist
3449:. 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
2160:{\displaystyle -3\zeta (-1)=\eta (-1)=\lim _{x\to 1^{-}}\left(1-2x+3x^{2}-4x^{3}+\cdots \right)=\lim _{x\to 1^{-}}{\frac {1}{(1+x)^{2}}}={\frac {1}{4}}.}
269:
where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an
3530:
video on the series, which gathered over 1.5 million views in its first month. The 8-minute video is narrated by Tony
Padilla, a physicist at the
425:
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
921:
4954:
2900:
640:, which is 4 times the original series. These relationships can be expressed using algebra. Whatever the "sum" of the series might be, call it
5301:
5394:
4295:
3791:
Pengelley, David J. (2002). "The bridge between the continuous and the discrete via original sources". In Otto Bekken; et al. (eds.).
636:, 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
4822:
4073:
2354:
3395:
3383:
3914:
Promoting numbers to functions is identified as one of two broad classes of summation methods, including Abel and Borel summation, by
4910:
5235:
2557:
and not fall into the pitfalls of divergent series. ... I told him that the sum of an infinite number of terms of the series:
5245:
3883:"Translation with notes of Euler's paper: Remarks on a beautiful relation between direct as well as reciprocal power series"
336:
72:
2595:
4632:
4373:
5409:
5240:
5000:
4947:
4400:"Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series"
3896:
Euler, Leonhard (1768). "Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques".
3800:
3775:
206:
3391:
the constant term −1/12, and the negative sign of this result reflects the fact that the
Casimir force is attractive.
1304:
does not converge. The benefit of introducing the
Riemann zeta function is that it can be defined for other values of
1066:
law alone. For an extreme example, appending a single zero to the front of the series can lead to a different result.
5389:
4756:
4728:
4697:
4495:
4246:
4181:
4024:
3997:
3961:
3936:
5399:
3711:
Lepowsky, J. (1999). "Vertex operator algebras and the zeta function". In
Naihuan Jing and Kailash C. Misra (ed.).
3001:
1849:
3829:
1349:
One method, along the lines of Euler's reasoning, uses the relationship between the
Riemann zeta function and the
5291:
5281:
3484:(−1), and they take the "lunatic asylum" line in his second letter as a sign that Ramanujan is toying with them.
2550:
2308:
621:
495:
4544:
609:" in chapter 8 of his first notebook. The simpler, less rigorous derivation proceeds in two steps, as follows.
4071:
Natiello, Mario A.; Solari, Hernan
Gustavo (July 2015), "On the removal of infinities from divergent series",
1219:
5404:
5306:
4940:
4884:
3608:
commented: "This calculation is one of the best-kept secrets in math. No one on the outside knows about it."
173:
are used in mathematics to assign numerical values even to a divergent series. In particular, the methods of
5432:
4399:
4162:"Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists"
3067:
1172:
1094:
538:
174:
4876:
The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation
3690:
The Euler–Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation
3135:
2298:. Instead, the method operates directly on conservative transformations of the series, using methods from
5414:
4166:
Quantum Field Theory I: Basics in
Mathematics and Physics. A Bridge Between Mathematicians and Physicists
3336:
2585:
2295:
3256:
3197:
1945:(−1) is an easier task, as the eta function is equal to the Abel sum of its defining series, which is a
1178:
5458:
5296:
4594:
4576:
3014:
445:
are used to assign numerical values to divergent series, some more powerful than others. For example,
5453:
5286:
5266:
3455:
3445:
3372:
2995:
4214:
3989:
3928:
3531:
915:, and then differentiating and negating both sides of the equation.) Accordingly, Ramanujan writes
421:
The infinite sequence of triangular numbers diverges to +∞, so by definition, the infinite series
5381:
5203:
4794:
Watson, G. N. (April 1929), "Theorems stated by
Ramanujan (VIII): Theorems on Divergent Series",
3477:
1918:
continues to hold when both functions are extended by analytic continuation to include values of
296:
mathematician Terry Gannon calls this equation "one of the most remarkable formulae in science".
4306:
3429:, Euler's early work on divergent series relied on function expansions, from which he concluded
2783:{\displaystyle c=-{\frac {1}{2}}f(0)-\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k)!}}f^{(2k-1)}(0),}
572:
5043:
4990:
4209:
3612:
1350:
4161:
3618:
describes the Numberphile video as misleading and notes that the interpretation of the sum as
2290:. Smoothing is a conceptual bridge between zeta function regularization, with its reliance on
5250:
4995:
4014:
3642:
3492:
3375:, which leads to bosonic string theory failing to be consistent in dimensions other than 26.
3325:
1309:
1279:
293:
3981:
3920:
2423:
is a cutoff function with appropriate properties. The cutoff function must be normalized to
5361:
5198:
4967:
4782:
4169:
4109:
3726:
278:
163:
3795:. National Center for Mathematics Education, University of Gothenburg, Sweden. p. 3.
8:
5341:
5208:
4826:
4685:
3982:
3921:
2515:
2443:
584:
577:
542:
178:
159:
4786:
4173:
4113:
3730:
446:
5271:
5182:
5167:
5139:
5119:
5058:
4849:
4772:
4549:
4365:
4347:
4269:
4227:
3882:
3746:
3716:
3600:
617:
450:
4416:
3768:
Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics
5371:
5172:
5144:
5098:
5088:
5068:
5053:
4926:
4752:
4724:
4693:
4491:
4451:
4177:
4121:
4020:
3993:
3957:
3932:
3796:
3771:
1063:
415:
313:
64:
4904:
4608:
4512:
4369:
5356:
5177:
5103:
5093:
5073:
4975:
4803:
4447:
4411:
4357:
4261:
4219:
4117:
4082:
3472:
2820:
2439:
2291:
1271:
442:
289:
274:
170:
56:
4438:
Barbeau, E. J.; Leah, P. J. (May 1976), "Euler's 1760 paper on divergent series",
3480:
discuss the meaning of this series. They conclude that Ramanujan has rediscovered
647:
Then multiply this equation by 4 and subtract the second equation from the first:
5134:
5063:
4485:
3654:
3653:. The Numberphile video was critiqued on similar grounds by German mathematician
3547:
3340:
2973:
To avoid inconsistencies, the modern theory of Ramanujan summation requires that
2435:
2268:
1946:
270:
4100:
Barbiellini, Bernardo (1987), "The Casimir effect in conformal field theories",
5366:
5351:
5346:
5025:
5010:
4919:
4863:
4540:
3605:
3487:
474:
304:
47:
4915:
4898:
4893:
4807:
4657:
4588:
4570:
273:
in its usual meaning. These methods have applications in other fields such as
5447:
5331:
5005:
4844:
4840:
4836:
4832:
4465:
3467:
2299:
282:
4888:
4361:
5336:
5078:
5020:
4197:
3977:
3916:
3426:
3387:
3347:
is the dimension of spacetime. If the fundamental oscillation frequency is
865:
411:
4769:
Proceedings of the II International Conference on Fundamental Interactions
1289:). On the other hand, the Dirichlet series diverges when the real part of
1017:{\displaystyle -3c=1-2+3-4+\cdots ={\frac {1}{(1+1)^{2}}}={\frac {1}{4}}.}
5083:
5030:
4879:
4707:
3684:
3523:
2963:{\displaystyle c=-{\frac {1}{6}}\times {\frac {1}{2!}}=-{\frac {1}{12}}.}
2542:
36:
4683:
4333:"A variation of Euler's approach to values of the Riemann zeta function"
3853:
Toils and triumphs of Srinivasa Ramanujan, the man and the mathematician
3661:
YouTube channel in 2018, his video receiving 2.7 million views by 2023.
2487:: it is necessarily the same value given by analytic continuation,
4932:
4273:
4231:
4086:
2447:
4777:
1278:
is greater than 1, the Dirichlet series converges, and its sum is the
1093:
can be set to −1 later. The implementation of this strategy is called
42:
is −1/8, and the area of the parabola underneath the y-axis is -1/12.
5015:
4859:
4853:
4352:
3721:
426:
32:
4265:
4223:
2584:
Ramanujan summation is a method to isolate the constant term in the
4963:
4767:
Elizalde, Emilio (2004). "Cosmology: Techniques and Applications".
2483:. The constant term of the asymptotic expansion does not depend on
2409:{\displaystyle \sum _{n=0}^{\infty }nf\left({\frac {n}{N}}\right),}
2236:
414:
as early as the sixth century BCE. Numbers of this form are called
155:
151:
28:
3751:
3745:
Tong, David (February 23, 2012). "String Theory". pp. 28–48.
2223:
3885:. Translated by Willis, Lucas; Osler, Thomas J. The Euler Archive
3713:
Recent Developments in Quantum Affine Algebras and Related Topics
3527:
3394:
A similar calculation is involved in three dimensions, using the
3359:/2. So using the divergent series, the sum over all harmonics is
19:
4331:
Kaneko, Masanobu; Kurokawa, Nobushige; Wakayama, Masato (2003),
477:
is a more powerful method that not only sums Grandi's series to
441:
is relatively difficult to manipulate into a finite value. Many
2977:
is "regular" in the sense that the higher-order derivatives of
2208:
3954:
A Primer of Analytic Number Theory: From Pythagoras to Riemann
3715:. Contemporary Mathematics. Vol. 248. pp. 327–340.
3312:
Therefore, every method that gives a finite value to the sum
2989:
612:
The first key insight is that the series of positive numbers
4633:"The Great Debate Over Whether 1 + 2 + 3 + 4... + ∞ = −1/12"
1105:
4305:(in French) (31), IREM de Strasbourg: 15–25, archived from
418:, because they can be arranged as an equilateral triangle.
3406:
It is unclear whether Leonhard Euler summed the series to
1293:
is less than or equal to 1, so, in particular, the series
4894:
Link to Numberphile video 1 + 2 + 3 + 4 + 5 + ... = –1/12
580:'s first notebook describing the "constant" of the series
4141:
3453:. In the same publication, Euler writes that the sum of
4899:
Sum of Natural Numbers (second proof and extra footage)
4590:
Sum of Natural Numbers (second proof and extra footage)
3351:, then the energy in an oscillator contributing to the
2305:
The idea is to replace the ill-behaved discrete series
4330:
2599:
2312:
1222:
1181:
860:
The second key insight is that the alternating series
564:
using some rough heuristics related to these methods.
5312:
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
5302:
1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
4038:
4036:
3259:
3200:
3138:
3070:
3017:
2903:
2653:
2598:
2357:
2311:
1958:
1852:
1369:
924:
656:
400:{\displaystyle \sum _{k=1}^{n}k={\frac {n(n+1)}{2}}.}
339:
209:
136:{\displaystyle \sum _{k=1}^{n}k={\frac {n(n+1)}{2}},}
75:
4833:
This Week's Finds in Mathematical Physics (Week 124)
4659:
Numberphile v. Math: the truth about 1+2+3+...=-1/12
4052:
2637:{\displaystyle \textstyle \sum _{k=1}^{\infty }f(k)}
2294:, and Ramanujan summation, with its shortcut to the
1323:
From this point, there are a few ways to prove that
911:
to the alternating sum of the nonnegative powers of
4885:
A recursive evaluation of zeta of negative integers
4136:
v:Quantum mechanics/Casimir effect in one dimension
4033:
4013:Aiyangar, Srinivasa Ramanujan (7 September 1995).
3810:
3298:
3239:
3180:
3118:
3050:
2962:
2782:
2636:
2408:
2340:
2159:
1910:
1835:
1312:. One can then define the zeta-regularized sum of
1262:
1208:
1016:
849:
399:
258:
135:
4905:What do we get if we sum all the natural numbers?
4610:What do we get if we sum all the natural numbers?
3604:coverage of the Numberphile video, mathematician
2588:for the partial sums of a series. For a function
1922:for which the above series diverge. Substituting
259:{\displaystyle 1+2+3+4+\cdots =-{\frac {1}{12}},}
5445:
4907:response to comments about video by Tony Padilla
4404:Proceedings of the American Mathematical Society
3371:. Ultimately it is this fact, combined with the
2088:
2002:
1142:leads to a region of negative values, including
545:. It is also possible to argue for the value of
4535:
4533:
2446:. One can then prove that this smoothed sum is
1100:
4200:(November 1983), "Euler and Infinite Series",
4070:
537:. More advanced methods are required, such as
4948:
3898:Mémoires de l'Académie des Sciences de Berlin
3793:Study the Masters: The Abel-Fauvel Conference
1911:{\displaystyle (1-2^{1-s})\zeta (s)=\eta (s)}
330:th partial sum is given by a simple formula:
5395:Hypergeometric function of a matrix argument
4530:
2592:, the classical Ramanujan sum of the series
892:defined as 1. (This can be seen by equating
5251:1 + 1/2 + 1/3 + ... (Riemann zeta function)
4927:Divergent Series: why 1 + 2 + 3 + ⋯ = −1/12
4572:ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = –1/12
4437:
4099:
4074:Philosophy of Mathematics Education Journal
3770:, Cambridge University Press, p. 140,
3438:
2341:{\displaystyle \textstyle \sum _{n=0}^{N}n}
1135:. Analytic continuation around the pole at
4955:
4941:
4796:Journal of the London Mathematical Society
4483:
3250:and subtracting the last two series gives
2990:Failure of stable linear summation methods
2541:. Ramanujan wrote in his second letter to
2240:Asymptotic behavior of the smoothing. The
200:, which is expressed by a famous formula:
23:The first four partial sums of the series
5307:1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
4929:by Brydon Cais from University of Arizona
4901:includes demonstration of Euler's method.
4858:
4776:
4415:
4351:
4213:
3984:Theory and Application of Infinite Series
3923:Theory and Application of Infinite Series
3790:
3750:
3720:
3443:alongside the divergent geometric series
3437:is mentioned in Euler's 1760 publication
1263:{\textstyle \sum _{n=1}^{\infty }n^{-s}.}
4962:
4850:Euler's Proof That 1 + 2 + 3 + ⋯ = −1/12
4766:
4746:
4012:
3710:
3637:relies on a specialized meaning for the
2894:is 1, and every other term vanishes, so
2235:
2198:
1104:
571:
303:
46:The infinite series whose terms are the
18:
4630:
4539:
4464:
4159:
3951:
3398:in place of the Riemann zeta function.
5446:
4793:
4510:
4397:
4042:
3865:
3765:
2509:
437:Among the classical divergent series,
4936:
4706:
4656:Polster, Burkard (January 13, 2018).
4545:"In the End, It All Adds Up to –1/12"
4244:
4196:
4058:
3976:
3915:
3895:
3880:
3816:
3679:
3677:
3675:
3673:
3119:{\displaystyle 0+1+2+3+\cdots =0+x=x}
2271:can "smooth" the series to arrive at
2267:The method of regularization using a
1270:The latter series is an example of a
3850:
3744:
3181:{\displaystyle 1+1+1+\cdots =x-x=0.}
2170:Dividing both sides by −3, one gets
1027:Dividing both sides by −3, one gets
493:, but also sums the trickier series
63:th partial sum of the series is the
5272:1 − 1 + 1 − 1 + ⋯ (Grandi's series)
4911:Related article from New York Times
4718:
4606:
4511:Thomas, Rachel (December 1, 2008),
4296:"Les séries divergentes chez Euler"
4147:
4047:, Springer-Verlag, pp. 13, 134
3870:, Springer-Verlag, pp. 135–136
3683:
3550:, who describes in more detail how
3462:
3191:Adding 0 to both sides again gives
1209:{\textstyle \sum _{n=1}^{\infty }n}
13:
4739:
4721:Quantum field theory in a nutshell
4398:Sondow, Jonathan (February 1994),
4293:
3670:
3382:is also involved in computing the
3299:{\displaystyle 1+0+0+0+\cdots =0,}
3240:{\displaystyle 0+1+1+1+\cdots =0,}
3061:then adding 0 to both sides gives
2704:
2616:
2374:
1239:
1198:
14:
5470:
5390:Generalized hypergeometric series
4918:follow-up Numberphile video with
4823:Does 1+2+3... Really Equal –1/12?
4815:
4692:. American Mathematical Society.
4690:Ramanujan: letters and commentary
4417:10.1090/S0002-9939-1994-1172954-7
4254:The American Mathematical Monthly
4016:Ramanujan: Letters and Commentary
3476:includes a scene where Hardy and
628:In order to transform the series
449:is a well-known method that sums
146:which increases without bound as
5428:
5427:
5400:Lauricella hypergeometric series
5118:
4245:Ayoub, Raymond (December 1974),
3051:{\displaystyle 1+2+3+\cdots =x,}
2801: − 1)th derivative of
2222:
2207:
308:The first six triangular numbers
5410:Riemann's differential equation
4749:A First Course in String Theory
4676:
4649:
4624:
4600:
4582:
4564:
4504:
4477:
4458:
4431:
4391:
4324:
4287:
4238:
4190:
4153:
4128:
4093:
4064:
4006:
3970:
3945:
3908:
3874:
531:
410:This equation was known to the
318:The partial sums of the series
299:
3859:
3844:
3822:
3784:
3759:
3738:
3704:
2774:
2768:
2763:
2748:
2734:
2725:
2682:
2676:
2630:
2624:
2479:is a constant that depends on
2244:-intercept of the parabola is
2129:
2116:
2095:
2009:
1995:
1986:
1977:
1968:
1905:
1899:
1890:
1884:
1878:
1853:
1823:
1817:
1682:
1676:
1541:
1535:
1383:
1377:
1077:is just a number. If the term
986:
973:
587:presented two derivations of "
453:, the mildly divergent series
432:
385:
373:
121:
109:
1:
5405:Modular hypergeometric series
5246:1/4 + 1/16 + 1/64 + 1/256 + ⋯
4631:Schultz, Colin (2014-01-31).
4340:Kyushu Journal of Mathematics
4247:"Euler and the Zeta Function"
4045:Ramanujan's Notebooks: Part 1
3868:Ramanujan's Notebooks: Part 1
3664:
3641:sign, from the techniques of
3316:is not stable or not linear.
567:
181:assign the series a value of
4688:; Rankin, Robert A. (1995).
4686:Srinivasa Ramanujan Aiyangar
4472:, Bloomsbury, pp. 61–62
4452:10.1016/0315-0860(76)90030-6
4122:10.1016/0370-2693(87)90854-9
3766:Gannon, Terry (April 2010),
3337:quantum harmonic oscillators
1173:zeta function regularization
1101:Zeta function regularization
1095:zeta function regularization
539:zeta function regularization
175:zeta function regularization
7:
5415:Theta hypergeometric series
3000:A summation method that is
1124:, the series converges and
528:
10:
5475:
5297:Infinite arithmetic series
5241:1/2 + 1/4 + 1/8 + 1/16 + ⋯
5236:1/2 − 1/4 + 1/8 − 1/16 + ⋯
4916:Why –1/12 is a gold nugget
4160:Zeidler, Eberhard (2007),
3851:Abdi, Wazir Hasan (1992),
3611:Coverage of this topic in
3542:and relates the latter to
3401:
3319:
2993:
2890:, the first derivative of
2545:, dated 27 February 1913:
1297:that results from setting
1216:is replaced by the series
1081:is promoted to a function
868:expansion of the function
311:
160:converge to a finite limit
5423:
5380:
5324:
5259:
5228:
5221:
5191:
5160:
5153:
5127:
5116:
5039:
4983:
4974:
4864:"My Favorite Numbers: 24"
4747:Zwiebach, Barton (2004).
4484:Complicite (April 2012),
4043:Berndt, Bruce C. (1985),
3952:Stopple, Jeffrey (2003),
3866:Berndt, Bruce C. (1985),
3440:De seriebus divergentibus
3309:contradicting stability.
518:Unlike the above series,
320:1 + 2 + 3 + 4 + 5 + 6 + ⋯
158:of partial sums fails to
3894:Originally published as
3881:Euler, Leonhard (2006).
3532:University of Nottingham
2348:with a smoothed version
1274:. When the real part of
1062:is not justified by the
5128:Properties of sequences
4808:10.1112/jlms/s1-4.14.82
4513:"A disappearing number"
4362:10.2206/kyushujm.57.175
2586:Euler–Maclaurin formula
2296:Euler–Maclaurin formula
4991:Arithmetic progression
4862:(September 19, 2008).
3855:, National, p. 41
3534:. Padilla begins with
3439:
3378:The regularization of
3300:
3241:
3182:
3120:
3052:
3004:cannot sum the series
2964:
2784:
2708:
2638:
2620:
2410:
2378:
2342:
2333:
2264:
2161:
1912:
1837:
1351:Dirichlet eta function
1264:
1243:
1210:
1202:
1168:
1018:
851:
616:closely resembles the
581:
523:and linear cannot sum
401:
360:
309:
260:
137:
96:
43:
5382:Hypergeometric series
4996:Geometric progression
4487:A Disappearing Number
4168:, Springer: 305–306,
3831:Ramanujan's Notebooks
3643:analytic continuation
3493:A Disappearing Number
3431:1 + 2 + 3 + 4 + ⋯ = ∞
3396:Epstein zeta-function
3373:Goddard–Thorn theorem
3326:bosonic string theory
3301:
3242:
3183:
3121:
3053:
2965:
2876:, and so on. Setting
2785:
2688:
2639:
2600:
2411:
2358:
2343:
2313:
2239:
2199:Cutoff regularization
2162:
1913:
1838:
1310:analytic continuation
1280:Riemann zeta function
1265:
1223:
1211:
1182:
1108:
1019:
852:
575:
402:
340:
307:
294:University of Alberta
261:
166:does not have a sum.
138:
76:
22:
5362:Trigonometric series
5154:Properties of series
5001:Harmonic progression
4669:– via YouTube.
4543:(February 3, 2014),
4440:Historia Mathematica
4312:on February 22, 2014
4202:Mathematics Magazine
3571:1 + 2 + 3 + 4 + ⋯ =
3569:as an Abel sum, and
3552:1 − 2 + 3 − 4 + ⋯ =
3499:1 + 2 + 3 + 4 + ⋯ =
3341:transverse direction
3257:
3198:
3136:
3068:
3015:
2901:
2651:
2596:
2559:1 + 2 + 3 + 4 + ⋯ =
2355:
2309:
1956:
1850:
1367:
1220:
1179:
922:
654:
645:= 1 + 2 + 3 + 4 + ⋯.
589:1 + 2 + 3 + 4 + ⋯ =
337:
279:quantum field theory
207:
73:
5342:Formal power series
4827:Scientific American
4787:2004gr.qc.....9076E
4174:2006qftb.book.....Z
4114:1987PhLB..190..137B
3731:1999math......9178L
2510:Ramanujan summation
2444:compactly supported
1060:= 0 + 4 + 0 + 8 + ⋯
638:4 + 8 + 12 + 16 + ⋯
585:Srinivasa Ramanujan
543:Ramanujan summation
179:Ramanujan summation
5140:Monotonic function
5059:Fibonacci sequence
4714:. Clarendon Press.
4684:Berndt, Bruce C.;
4550:The New York Times
3988:. Dover. pp.
3927:. Dover. pp.
3687:(April 10, 2010),
3651:is associated with
3601:The New York Times
3296:
3237:
3178:
3116:
3048:
2960:
2780:
2634:
2633:
2406:
2338:
2337:
2265:
2157:
2109:
2023:
1908:
1833:
1831:
1260:
1206:
1169:
1014:
847:
845:
618:alternating series
582:
527:to a finite value
416:triangular numbers
397:
310:
288:In a monograph on
256:
133:
44:
31:is their smoothed
5459:Arithmetic series
5441:
5440:
5372:Generating series
5320:
5319:
5292:1 − 2 + 4 − 8 + ⋯
5287:1 + 2 + 4 + 8 + ⋯
5282:1 − 2 + 3 − 4 + ⋯
5277:1 + 2 + 3 + 4 + ⋯
5267:1 + 1 + 1 + 1 + ⋯
5217:
5216:
5145:Periodic sequence
5114:
5113:
5099:Triangular number
5089:Pentagonal number
5069:Heptagonal number
5054:Complete sequence
4976:Integer sequences
4821:Lamb E. (2014), "
4260:(10): 1067–1086,
4102:Physics Letters B
3544:1 + 2 + 3 + 4 + ⋯
3540:1 − 2 + 3 − 4 + ⋯
3536:1 − 1 + 1 − 1 + ⋯
3522:In January 2014,
3456:1 + 1 + 1 + 1 + ⋯
3451:1 + 2 + 3 + 4 + ⋯
3446:1 + 2 + 4 + 8 + ⋯
3435:1 + 2 + 3 + 4 + ⋯
3380:1 + 2 + 3 + 4 + ⋯
3002:linear and stable
2996:1 + 1 + 1 + 1 + ⋯
2984:0 + 2 + 0 + 4 + ⋯
2955:
2939:
2921:
2741:
2671:
2520:1 + 2 + 3 + 4 + ⋯
2397:
2216:1 + 2 + 3 + 4 + ⋯
2152:
2139:
2087:
2001:
1941:. Now, computing
1314:1 + 2 + 3 + 4 + ⋯
1295:1 + 2 + 3 + 4 + ⋯
1071:1 + 2 + 3 + 4 + ⋯
1064:additive identity
1009:
996:
862:1 − 2 + 3 − 4 + ⋯
634:1 − 2 + 3 − 4 + ⋯
630:1 + 2 + 3 + 4 + ⋯
622:1 − 2 + 3 − 4 + ⋯
614:1 + 2 + 3 + 4 + ⋯
532:§ Heuristics
520:1 + 2 + 3 + 4 + ⋯
496:1 − 2 + 3 − 4 + ⋯
455:1 − 1 + 1 − 1 + ⋯
443:summation methods
439:1 + 2 + 3 + 4 + ⋯
423:1 + 2 + 3 + 4 + ⋯
392:
314:Triangular number
251:
171:summation methods
128:
65:triangular number
52:1 + 2 + 3 + 4 + ⋯
25:1 + 2 + 3 + 4 + ⋯
5466:
5454:Divergent series
5431:
5430:
5357:Dirichlet series
5226:
5225:
5158:
5157:
5122:
5094:Polygonal number
5074:Hexagonal number
5047:
4981:
4980:
4957:
4950:
4943:
4934:
4933:
4870:
4868:
4810:
4790:
4780:
4763:See p. 293.
4762:
4751:. Cambridge UP.
4734:
4723:. Princeton UP.
4719:Zee, A. (2003).
4715:
4712:Divergent Series
4703:
4671:
4670:
4668:
4666:
4653:
4647:
4646:
4644:
4643:
4628:
4622:
4620:
4619:
4617:
4604:
4598:
4591:
4586:
4580:
4573:
4568:
4562:
4560:
4559:
4557:
4537:
4528:
4526:
4525:
4523:
4508:
4502:
4500:
4481:
4475:
4473:
4470:The Indian Clerk
4462:
4456:
4454:
4435:
4429:
4427:
4426:
4424:
4419:
4395:
4389:
4387:
4386:
4384:
4378:
4372:, archived from
4355:
4337:
4328:
4322:
4320:
4319:
4317:
4311:
4300:
4291:
4285:
4283:
4282:
4280:
4251:
4242:
4236:
4234:
4217:
4194:
4188:
4186:
4157:
4151:
4145:
4139:
4132:
4126:
4124:
4108:(1–2): 137–139,
4097:
4091:
4089:
4068:
4062:
4056:
4050:
4048:
4040:
4031:
4030:
4010:
4004:
4003:
3987:
3974:
3968:
3966:
3949:
3943:
3942:
3926:
3912:
3906:
3905:
3893:
3891:
3890:
3878:
3872:
3871:
3863:
3857:
3856:
3848:
3842:
3841:
3840:
3838:
3826:
3820:
3814:
3808:
3806:
3788:
3782:
3780:
3763:
3757:
3756:
3754:
3742:
3736:
3734:
3724:
3708:
3702:
3700:
3699:
3697:
3681:
3636:
3634:
3633:
3630:
3627:
3623:
3590:
3589:
3587:
3586:
3583:
3580:
3576:
3568:
3567:
3565:
3564:
3561:
3558:
3545:
3541:
3537:
3518:
3517:
3515:
3514:
3511:
3508:
3504:
3473:The Indian Clerk
3463:In popular media
3458:
3452:
3448:
3442:
3436:
3432:
3424:
3422:
3421:
3418:
3415:
3411:
3381:
3370:
3334:
3315:
3305:
3303:
3302:
3297:
3246:
3244:
3243:
3238:
3187:
3185:
3184:
3179:
3125:
3123:
3122:
3117:
3057:
3055:
3054:
3049:
3007:
2985:
2969:
2967:
2966:
2961:
2956:
2948:
2940:
2938:
2927:
2922:
2914:
2889:
2875:
2874:
2872:
2871:
2868:
2865:
2861:
2847:
2846:
2844:
2843:
2840:
2837:
2821:Bernoulli number
2789:
2787:
2786:
2781:
2767:
2766:
2742:
2740:
2723:
2722:
2710:
2707:
2702:
2672:
2664:
2643:
2641:
2640:
2635:
2619:
2614:
2578:
2577:
2575:
2574:
2571:
2568:
2564:
2540:
2538:
2537:
2534:
2531:
2527:
2521:
2505:
2503:
2502:
2499:
2496:
2492:
2474:
2469:
2467:
2466:
2463:
2460:
2456:
2429:
2415:
2413:
2412:
2407:
2402:
2398:
2390:
2377:
2372:
2347:
2345:
2344:
2339:
2332:
2327:
2292:complex analysis
2289:
2287:
2286:
2283:
2280:
2276:
2262:
2260:
2259:
2256:
2253:
2249:
2226:
2217:
2211:
2195:
2193:
2191:
2190:
2187:
2184:
2180:
2166:
2164:
2163:
2158:
2153:
2145:
2140:
2138:
2137:
2136:
2111:
2108:
2107:
2106:
2083:
2079:
2072:
2071:
2056:
2055:
2022:
2021:
2020:
1940:
1928:
1917:
1915:
1914:
1909:
1877:
1876:
1842:
1840:
1839:
1834:
1832:
1800:
1799:
1784:
1783:
1768:
1765:
1763:
1762:
1747:
1746:
1731:
1728:
1726:
1725:
1710:
1709:
1695:
1690:
1672:
1668:
1667:
1666:
1637:
1629:
1628:
1607:
1604:
1602:
1601:
1580:
1577:
1575:
1574:
1554:
1549:
1531:
1530:
1509:
1501:
1500:
1485:
1484:
1469:
1466:
1464:
1463:
1448:
1447:
1432:
1429:
1427:
1426:
1411:
1410:
1396:
1391:
1348:
1346:
1344:
1343:
1340:
1337:
1333:
1315:
1303:
1296:
1272:Dirichlet series
1269:
1267:
1266:
1261:
1256:
1255:
1242:
1237:
1215:
1213:
1212:
1207:
1201:
1196:
1166:
1165:
1163:
1162:
1159:
1156:
1152:
1141:
1134:
1123:
1072:
1061:
1049:
1047:
1046:
1043:
1040:
1036:
1023:
1021:
1020:
1015:
1010:
1002:
997:
995:
994:
993:
968:
910:
908:
907:
901:
898:
887:
885:
884:
877:
874:
863:
856:
854:
853:
848:
846:
824:
821:
807:
804:
792:
790:
757:
754:
746:
743:
737:
735:
702:
699:
685:
682:
670:
668:
646:
639:
635:
631:
624:
615:
608:
607:
605:
604:
601:
598:
594:
563:
561:
560:
557:
554:
550:
536:
526:
521:
514:
512:
511:
508:
505:
498:
492:
490:
489:
486:
483:
472:
470:
469:
466:
463:
456:
447:Cesàro summation
440:
424:
406:
404:
403:
398:
393:
388:
368:
359:
354:
325:
321:
290:moonshine theory
275:complex analysis
265:
263:
262:
257:
252:
244:
199:
197:
196:
193:
190:
186:
142:
140:
139:
134:
129:
124:
104:
95:
90:
57:divergent series
53:
26:
16:Divergent series
5474:
5473:
5469:
5468:
5467:
5465:
5464:
5463:
5444:
5443:
5442:
5437:
5419:
5376:
5325:Kinds of series
5316:
5255:
5222:Explicit series
5213:
5187:
5149:
5135:Cauchy sequence
5123:
5110:
5064:Figurate number
5041:
5035:
5026:Powers of three
4970:
4961:
4866:
4818:
4813:
4759:
4742:
4740:Further reading
4737:
4731:
4700:
4679:
4674:
4664:
4662:
4655:
4654:
4650:
4641:
4639:
4629:
4625:
4615:
4613:
4607:Padilla, Tony,
4605:
4601:
4589:
4587:
4583:
4571:
4569:
4565:
4555:
4553:
4541:Overbye, Dennis
4538:
4531:
4521:
4519:
4509:
4505:
4498:
4482:
4478:
4463:
4459:
4436:
4432:
4422:
4420:
4396:
4392:
4382:
4380:
4376:
4335:
4329:
4325:
4315:
4313:
4309:
4298:
4292:
4288:
4278:
4276:
4266:10.2307/2319041
4249:
4243:
4239:
4224:10.2307/2690371
4215:10.1.1.639.6923
4195:
4191:
4184:
4158:
4154:
4146:
4142:
4133:
4129:
4098:
4094:
4069:
4065:
4057:
4053:
4041:
4034:
4027:
4011:
4007:
4000:
3975:
3971:
3964:
3956:, p. 202,
3950:
3946:
3939:
3913:
3909:
3888:
3886:
3879:
3875:
3864:
3860:
3849:
3845:
3836:
3834:
3828:
3827:
3823:
3815:
3811:
3803:
3789:
3785:
3778:
3764:
3760:
3743:
3739:
3709:
3705:
3695:
3693:
3682:
3671:
3667:
3655:Burkard Polster
3631:
3628:
3625:
3624:
3621:
3619:
3584:
3581:
3578:
3577:
3574:
3572:
3570:
3562:
3559:
3556:
3555:
3553:
3551:
3543:
3539:
3535:
3512:
3509:
3506:
3505:
3502:
3500:
3498:
3465:
3454:
3450:
3444:
3434:
3430:
3425:. According to
3419:
3416:
3413:
3412:
3409:
3407:
3404:
3379:
3360:
3355:th harmonic is
3339:, one for each
3329:
3322:
3313:
3258:
3255:
3254:
3199:
3196:
3195:
3137:
3134:
3133:
3069:
3066:
3065:
3016:
3013:
3012:
3005:
2998:
2992:
2983:
2947:
2931:
2926:
2913:
2902:
2899:
2898:
2877:
2869:
2866:
2863:
2862:
2859:
2857:
2855:
2849:
2841:
2838:
2835:
2834:
2832:
2830:
2824:
2814:
2747:
2743:
2724:
2715:
2711:
2709:
2703:
2692:
2663:
2652:
2649:
2648:
2615:
2604:
2597:
2594:
2593:
2572:
2569:
2566:
2565:
2562:
2560:
2558:
2555:Infinite Series
2535:
2532:
2529:
2528:
2525:
2523:
2519:
2512:
2500:
2497:
2494:
2493:
2490:
2488:
2464:
2461:
2458:
2457:
2454:
2452:
2451:
2424:
2389:
2385:
2373:
2362:
2356:
2353:
2352:
2328:
2317:
2310:
2307:
2306:
2284:
2281:
2278:
2277:
2274:
2272:
2269:cutoff function
2257:
2254:
2251:
2250:
2247:
2245:
2234:
2233:
2232:
2231:
2230:
2229:After smoothing
2227:
2219:
2218:
2215:
2212:
2201:
2188:
2185:
2182:
2181:
2178:
2176:
2171:
2144:
2132:
2128:
2115:
2110:
2102:
2098:
2091:
2067:
2063:
2051:
2047:
2028:
2024:
2016:
2012:
2005:
1957:
1954:
1953:
1947:one-sided limit
1930:
1923:
1866:
1862:
1851:
1848:
1847:
1830:
1829:
1807:
1792:
1788:
1776:
1772:
1767:
1764:
1755:
1751:
1739:
1735:
1730:
1727:
1718:
1714:
1702:
1698:
1696:
1694:
1689:
1685:
1656:
1652:
1645:
1641:
1638:
1636:
1621:
1617:
1606:
1603:
1594:
1590:
1579:
1576:
1567:
1563:
1555:
1553:
1548:
1544:
1523:
1519:
1510:
1508:
1493:
1489:
1477:
1473:
1468:
1465:
1456:
1452:
1440:
1436:
1431:
1428:
1419:
1415:
1403:
1399:
1397:
1395:
1390:
1386:
1370:
1368:
1365:
1364:
1341:
1338:
1335:
1334:
1331:
1329:
1324:
1313:
1298:
1294:
1248:
1244:
1238:
1227:
1221:
1218:
1217:
1197:
1186:
1180:
1177:
1176:
1160:
1157:
1154:
1153:
1150:
1148:
1143:
1136:
1125:
1118:
1103:
1070:
1055:
1044:
1041:
1038:
1037:
1034:
1032:
1001:
989:
985:
972:
967:
923:
920:
919:
902:
899:
896:
895:
893:
878:
875:
872:
871:
869:
861:
844:
843:
823:
820:
806:
803:
791:
789:
771:
770:
756:
753:
745:
742:
736:
734:
722:
721:
701:
698:
684:
681:
669:
667:
657:
655:
652:
651:
641:
637:
633:
629:
620:
613:
602:
599:
596:
595:
592:
590:
588:
570:
558:
555:
552:
551:
548:
546:
524:
519:
509:
506:
503:
502:
500:
494:
487:
484:
481:
480:
478:
467:
464:
461:
460:
458:
454:
451:Grandi's series
438:
435:
422:
369:
367:
355:
344:
338:
335:
334:
324:1, 3, 6, 10, 15
323:
319:
316:
302:
271:infinite series
243:
208:
205:
204:
194:
191:
188:
187:
184:
182:
105:
103:
91:
80:
74:
71:
70:
51:
48:natural numbers
24:
17:
12:
11:
5:
5472:
5462:
5461:
5456:
5439:
5438:
5436:
5435:
5424:
5421:
5420:
5418:
5417:
5412:
5407:
5402:
5397:
5392:
5386:
5384:
5378:
5377:
5375:
5374:
5369:
5367:Fourier series
5364:
5359:
5354:
5352:Puiseux series
5349:
5347:Laurent series
5344:
5339:
5334:
5328:
5326:
5322:
5321:
5318:
5317:
5315:
5314:
5309:
5304:
5299:
5294:
5289:
5284:
5279:
5274:
5269:
5263:
5261:
5257:
5256:
5254:
5253:
5248:
5243:
5238:
5232:
5230:
5223:
5219:
5218:
5215:
5214:
5212:
5211:
5206:
5201:
5195:
5193:
5189:
5188:
5186:
5185:
5180:
5175:
5170:
5164:
5162:
5155:
5151:
5150:
5148:
5147:
5142:
5137:
5131:
5129:
5125:
5124:
5117:
5115:
5112:
5111:
5109:
5108:
5107:
5106:
5096:
5091:
5086:
5081:
5076:
5071:
5066:
5061:
5056:
5050:
5048:
5037:
5036:
5034:
5033:
5028:
5023:
5018:
5013:
5008:
5003:
4998:
4993:
4987:
4985:
4978:
4972:
4971:
4960:
4959:
4952:
4945:
4937:
4931:
4930:
4924:
4923:
4922:
4920:Edward Frenkel
4913:
4908:
4902:
4891:
4882:
4873:
4872:
4871:
4856:
4830:
4817:
4816:External links
4814:
4812:
4811:
4791:
4764:
4757:
4743:
4741:
4738:
4736:
4735:
4729:
4716:
4704:
4698:
4680:
4678:
4675:
4673:
4672:
4648:
4623:
4599:
4581:
4563:
4529:
4503:
4496:
4476:
4466:Leavitt, David
4457:
4446:(2): 141–160,
4430:
4410:(4): 421–424,
4390:
4346:(1): 175–192,
4323:
4294:Lefort, Jean,
4286:
4237:
4208:(5): 307–314,
4189:
4182:
4152:
4140:
4127:
4092:
4063:
4061:, p. 346.
4051:
4032:
4025:
4019:. p. 53.
4005:
3998:
3969:
3962:
3944:
3937:
3907:
3873:
3858:
3843:
3821:
3809:
3802:978-9185143009
3801:
3783:
3777:978-0521141888
3776:
3758:
3737:
3703:
3668:
3666:
3663:
3606:Edward Frenkel
3488:Simon McBurney
3470:'s 2007 novel
3464:
3461:
3403:
3400:
3321:
3318:
3307:
3306:
3295:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3265:
3262:
3248:
3247:
3236:
3233:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3206:
3203:
3189:
3188:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3127:
3126:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3059:
3058:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
2991:
2988:
2971:
2970:
2959:
2954:
2951:
2946:
2943:
2937:
2934:
2930:
2925:
2920:
2917:
2912:
2909:
2906:
2853:
2828:
2809:
2791:
2790:
2779:
2776:
2773:
2770:
2765:
2762:
2759:
2756:
2753:
2750:
2746:
2739:
2736:
2733:
2730:
2727:
2721:
2718:
2714:
2706:
2701:
2698:
2695:
2691:
2687:
2684:
2681:
2678:
2675:
2670:
2667:
2662:
2659:
2656:
2644:is defined as
2632:
2629:
2626:
2623:
2618:
2613:
2610:
2607:
2603:
2582:
2581:
2511:
2508:
2417:
2416:
2405:
2401:
2396:
2393:
2388:
2384:
2381:
2376:
2371:
2368:
2365:
2361:
2336:
2331:
2326:
2323:
2320:
2316:
2228:
2221:
2220:
2213:
2206:
2205:
2204:
2203:
2202:
2200:
2197:
2168:
2167:
2156:
2151:
2148:
2143:
2135:
2131:
2127:
2124:
2121:
2118:
2114:
2105:
2101:
2097:
2094:
2090:
2086:
2082:
2078:
2075:
2070:
2066:
2062:
2059:
2054:
2050:
2046:
2043:
2040:
2037:
2034:
2031:
2027:
2019:
2015:
2011:
2008:
2004:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1875:
1872:
1869:
1865:
1861:
1858:
1855:
1844:
1843:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1808:
1806:
1803:
1798:
1795:
1791:
1787:
1782:
1779:
1775:
1771:
1766:
1761:
1758:
1754:
1750:
1745:
1742:
1738:
1734:
1729:
1724:
1721:
1717:
1713:
1708:
1705:
1701:
1697:
1693:
1688:
1686:
1684:
1681:
1678:
1675:
1671:
1665:
1662:
1659:
1655:
1651:
1648:
1644:
1640:
1639:
1635:
1632:
1627:
1624:
1620:
1616:
1613:
1610:
1605:
1600:
1597:
1593:
1589:
1586:
1583:
1578:
1573:
1570:
1566:
1562:
1559:
1556:
1552:
1547:
1545:
1543:
1540:
1537:
1534:
1529:
1526:
1522:
1518:
1515:
1512:
1511:
1507:
1504:
1499:
1496:
1492:
1488:
1483:
1480:
1476:
1472:
1467:
1462:
1459:
1455:
1451:
1446:
1443:
1439:
1435:
1430:
1425:
1422:
1418:
1414:
1409:
1406:
1402:
1398:
1394:
1389:
1387:
1385:
1382:
1379:
1376:
1373:
1372:
1259:
1254:
1251:
1247:
1241:
1236:
1233:
1230:
1226:
1205:
1200:
1195:
1192:
1189:
1185:
1102:
1099:
1025:
1024:
1013:
1008:
1005:
1000:
992:
988:
984:
981:
978:
975:
971:
966:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
927:
864:is the formal
858:
857:
842:
839:
836:
833:
830:
827:
822:
819:
816:
813:
810:
805:
802:
799:
796:
793:
788:
785:
782:
779:
776:
773:
772:
769:
766:
763:
760:
755:
752:
749:
744:
741:
738:
733:
730:
727:
724:
723:
720:
717:
714:
711:
708:
705:
700:
697:
694:
691:
688:
683:
680:
677:
674:
671:
666:
663:
660:
659:
569:
566:
475:Abel summation
434:
431:
408:
407:
396:
391:
387:
384:
381:
378:
375:
372:
366:
363:
358:
353:
350:
347:
343:
312:Main article:
301:
298:
267:
266:
255:
250:
247:
242:
239:
236:
233:
230:
227:
224:
221:
218:
215:
212:
154:. Because the
144:
143:
132:
127:
123:
120:
117:
114:
111:
108:
102:
99:
94:
89:
86:
83:
79:
15:
9:
6:
4:
3:
2:
5471:
5460:
5457:
5455:
5452:
5451:
5449:
5434:
5426:
5425:
5422:
5416:
5413:
5411:
5408:
5406:
5403:
5401:
5398:
5396:
5393:
5391:
5388:
5387:
5385:
5383:
5379:
5373:
5370:
5368:
5365:
5363:
5360:
5358:
5355:
5353:
5350:
5348:
5345:
5343:
5340:
5338:
5335:
5333:
5332:Taylor series
5330:
5329:
5327:
5323:
5313:
5310:
5308:
5305:
5303:
5300:
5298:
5295:
5293:
5290:
5288:
5285:
5283:
5280:
5278:
5275:
5273:
5270:
5268:
5265:
5264:
5262:
5258:
5252:
5249:
5247:
5244:
5242:
5239:
5237:
5234:
5233:
5231:
5227:
5224:
5220:
5210:
5207:
5205:
5202:
5200:
5197:
5196:
5194:
5190:
5184:
5181:
5179:
5176:
5174:
5171:
5169:
5166:
5165:
5163:
5159:
5156:
5152:
5146:
5143:
5141:
5138:
5136:
5133:
5132:
5130:
5126:
5121:
5105:
5102:
5101:
5100:
5097:
5095:
5092:
5090:
5087:
5085:
5082:
5080:
5077:
5075:
5072:
5070:
5067:
5065:
5062:
5060:
5057:
5055:
5052:
5051:
5049:
5045:
5038:
5032:
5029:
5027:
5024:
5022:
5021:Powers of two
5019:
5017:
5014:
5012:
5009:
5007:
5006:Square number
5004:
5002:
4999:
4997:
4994:
4992:
4989:
4988:
4986:
4982:
4979:
4977:
4973:
4969:
4965:
4958:
4953:
4951:
4946:
4944:
4939:
4938:
4935:
4928:
4925:
4921:
4917:
4914:
4912:
4909:
4906:
4903:
4900:
4897:
4896:
4895:
4892:
4890:
4886:
4883:
4881:
4877:
4874:
4865:
4861:
4857:
4855:
4851:
4848:
4847:
4846:
4842:
4838:
4834:
4831:
4828:
4824:
4820:
4819:
4809:
4805:
4801:
4797:
4792:
4788:
4784:
4779:
4778:gr-qc/0409076
4774:
4770:
4765:
4760:
4758:0-521-83143-1
4754:
4750:
4745:
4744:
4732:
4730:0-691-01019-6
4726:
4722:
4717:
4713:
4709:
4705:
4701:
4699:0-8218-0287-9
4695:
4691:
4687:
4682:
4681:
4661:
4660:
4652:
4638:
4634:
4627:
4612:
4611:
4603:
4596:
4592:
4585:
4578:
4574:
4567:
4552:
4551:
4546:
4542:
4536:
4534:
4518:
4514:
4507:
4499:
4497:9781849432993
4493:
4489:
4488:
4480:
4471:
4467:
4461:
4453:
4449:
4445:
4441:
4434:
4418:
4413:
4409:
4405:
4401:
4394:
4379:on 2014-02-02
4375:
4371:
4367:
4363:
4359:
4354:
4349:
4345:
4341:
4334:
4327:
4308:
4304:
4297:
4290:
4275:
4271:
4267:
4263:
4259:
4255:
4248:
4241:
4233:
4229:
4225:
4221:
4216:
4211:
4207:
4203:
4199:
4198:Kline, Morris
4193:
4185:
4183:9783540347644
4179:
4175:
4171:
4167:
4163:
4156:
4149:
4144:
4137:
4131:
4123:
4119:
4115:
4111:
4107:
4103:
4096:
4088:
4084:
4080:
4076:
4075:
4067:
4060:
4055:
4046:
4039:
4037:
4028:
4026:9780821891254
4022:
4018:
4017:
4009:
4001:
3999:0-486-66165-2
3995:
3991:
3986:
3985:
3979:
3978:Knopp, Konrad
3973:
3965:
3963:0-521-81309-3
3959:
3955:
3948:
3940:
3938:0-486-66165-2
3934:
3930:
3925:
3924:
3918:
3917:Knopp, Konrad
3911:
3903:
3900:(in French).
3899:
3884:
3877:
3869:
3862:
3854:
3847:
3833:
3832:
3825:
3819:, p. 10.
3818:
3813:
3804:
3798:
3794:
3787:
3779:
3773:
3769:
3762:
3753:
3748:
3741:
3732:
3728:
3723:
3718:
3714:
3707:
3692:
3691:
3686:
3680:
3678:
3676:
3674:
3669:
3662:
3660:
3656:
3652:
3648:
3644:
3640:
3617:
3615:
3609:
3607:
3603:
3602:
3596:
3594:
3549:
3533:
3529:
3525:
3520:
3495:
3494:
3490:'s 2007 play
3489:
3485:
3483:
3479:
3475:
3474:
3469:
3468:David Leavitt
3460:
3459:is infinite.
3457:
3447:
3441:
3428:
3399:
3397:
3392:
3389:
3385:
3384:Casimir force
3376:
3374:
3368:
3364:
3358:
3354:
3350:
3346:
3342:
3338:
3332:
3327:
3317:
3314:1 + 2 + 3 + ⋯
3310:
3293:
3290:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3253:
3252:
3251:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3194:
3193:
3192:
3175:
3172:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3132:
3131:
3130:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3064:
3063:
3062:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3011:
3010:
3009:
3006:1 + 2 + 3 + ⋯
3003:
2997:
2987:
2980:
2976:
2957:
2952:
2949:
2944:
2941:
2935:
2932:
2928:
2923:
2918:
2915:
2910:
2907:
2904:
2897:
2896:
2895:
2893:
2888:
2884:
2880:
2852:
2827:
2822:
2818:
2813:
2808:
2804:
2800:
2796:
2777:
2771:
2760:
2757:
2754:
2751:
2744:
2737:
2731:
2728:
2719:
2716:
2712:
2699:
2696:
2693:
2689:
2685:
2679:
2673:
2668:
2665:
2660:
2657:
2654:
2647:
2646:
2645:
2627:
2621:
2611:
2608:
2605:
2601:
2591:
2587:
2556:
2552:
2548:
2547:
2546:
2544:
2522:is also
2517:
2516:Ramanujan sum
2507:
2486:
2482:
2478:
2473:
2449:
2445:
2441:
2437:
2433:
2427:
2422:
2403:
2399:
2394:
2391:
2386:
2382:
2379:
2369:
2366:
2363:
2359:
2351:
2350:
2349:
2334:
2329:
2324:
2321:
2318:
2314:
2303:
2301:
2300:real analysis
2297:
2293:
2270:
2243:
2238:
2225:
2210:
2196:
2174:
2154:
2149:
2146:
2141:
2133:
2125:
2122:
2119:
2112:
2103:
2099:
2092:
2084:
2080:
2076:
2073:
2068:
2064:
2060:
2057:
2052:
2048:
2044:
2041:
2038:
2035:
2032:
2029:
2025:
2017:
2013:
2006:
1998:
1992:
1989:
1983:
1980:
1974:
1971:
1965:
1962:
1959:
1952:
1951:
1950:
1948:
1944:
1938:
1934:
1926:
1921:
1902:
1896:
1893:
1887:
1881:
1873:
1870:
1867:
1863:
1859:
1856:
1846:The identity
1826:
1820:
1814:
1811:
1809:
1804:
1801:
1796:
1793:
1789:
1785:
1780:
1777:
1773:
1769:
1759:
1756:
1752:
1748:
1743:
1740:
1736:
1732:
1722:
1719:
1715:
1711:
1706:
1703:
1699:
1691:
1687:
1679:
1673:
1669:
1663:
1660:
1657:
1653:
1649:
1646:
1642:
1633:
1630:
1625:
1622:
1618:
1614:
1611:
1608:
1598:
1595:
1591:
1587:
1584:
1581:
1571:
1568:
1564:
1560:
1557:
1550:
1546:
1538:
1532:
1527:
1524:
1520:
1516:
1513:
1505:
1502:
1497:
1494:
1490:
1486:
1481:
1478:
1474:
1470:
1460:
1457:
1453:
1449:
1444:
1441:
1437:
1433:
1423:
1420:
1416:
1412:
1407:
1404:
1400:
1392:
1388:
1380:
1374:
1363:
1362:
1361:
1359:
1355:
1352:
1327:
1321:
1319:
1311:
1307:
1301:
1292:
1288:
1284:
1281:
1277:
1273:
1257:
1252:
1249:
1245:
1234:
1231:
1228:
1224:
1203:
1193:
1190:
1187:
1183:
1175:, the series
1174:
1146:
1139:
1132:
1128:
1121:
1116:
1112:
1107:
1098:
1096:
1092:
1088:
1084:
1080:
1076:
1067:
1065:
1059:
1051:
1031: =
1030:
1011:
1006:
1003:
998:
990:
982:
979:
976:
969:
964:
961:
958:
955:
952:
949:
946:
943:
940:
937:
934:
931:
928:
925:
918:
917:
916:
914:
906:
891:
882:
867:
840:
837:
834:
831:
828:
825:
817:
814:
811:
808:
800:
797:
794:
786:
783:
780:
777:
774:
767:
764:
761:
758:
750:
747:
739:
731:
728:
725:
718:
715:
712:
709:
706:
703:
695:
692:
689:
686:
678:
675:
672:
664:
661:
650:
649:
648:
644:
626:
623:
619:
610:
586:
579:
576:Passage from
574:
565:
544:
540:
534:
533:
525:1 + 2 + 3 + ⋯
516:
497:
476:
452:
448:
444:
430:
428:
419:
417:
413:
394:
389:
382:
379:
376:
370:
364:
361:
356:
351:
348:
345:
341:
333:
332:
331:
329:
315:
306:
297:
295:
291:
286:
284:
283:string theory
280:
276:
272:
253:
248:
245:
240:
237:
234:
231:
228:
225:
222:
219:
216:
213:
210:
203:
202:
201:
180:
176:
172:
167:
165:
161:
157:
153:
149:
130:
125:
118:
115:
112:
106:
100:
97:
92:
87:
84:
81:
77:
69:
68:
67:
66:
62:
58:
54:
49:
41:
39:
34:
30:
21:
5337:Power series
5276:
5079:Lucas number
5031:Powers of 10
5011:Cubic number
4802:(2): 82–86,
4799:
4795:
4768:
4748:
4720:
4711:
4708:Hardy, G. H.
4689:
4677:Bibliography
4663:. Retrieved
4658:
4651:
4640:. Retrieved
4636:
4626:
4614:, retrieved
4609:
4602:
4584:
4566:
4554:, retrieved
4548:
4520:, retrieved
4516:
4506:
4486:
4479:
4469:
4460:
4443:
4439:
4433:
4423:February 14,
4421:, retrieved
4407:
4403:
4393:
4381:, retrieved
4374:the original
4353:math/0206171
4343:
4339:
4326:
4316:February 14,
4314:, retrieved
4307:the original
4302:
4289:
4279:February 14,
4277:, retrieved
4257:
4253:
4240:
4205:
4201:
4192:
4165:
4155:
4150:, pp. 65–67.
4143:
4130:
4105:
4101:
4095:
4078:
4072:
4066:
4054:
4044:
4015:
4008:
3983:
3972:
3953:
3947:
3922:
3910:
3901:
3897:
3887:. Retrieved
3876:
3867:
3861:
3852:
3846:
3835:, retrieved
3830:
3824:
3812:
3792:
3786:
3767:
3761:
3740:
3722:math/9909178
3712:
3706:
3694:, retrieved
3689:
3685:Tao, Terence
3658:
3650:
3646:
3638:
3613:
3610:
3599:
3597:
3592:
3521:
3491:
3486:
3481:
3471:
3466:
3427:Morris Kline
3405:
3393:
3388:scalar field
3377:
3366:
3362:
3356:
3352:
3348:
3344:
3335:independent
3330:
3323:
3311:
3308:
3249:
3190:
3128:
3060:
2999:
2978:
2974:
2972:
2891:
2886:
2882:
2878:
2850:
2825:
2816:
2811:
2806:
2802:
2798:
2794:
2792:
2589:
2583:
2554:
2513:
2484:
2480:
2476:
2471:
2431:
2425:
2420:
2418:
2304:
2266:
2241:
2172:
2169:
1942:
1936:
1932:
1924:
1919:
1845:
1357:
1353:
1325:
1322:
1317:
1305:
1299:
1290:
1286:
1282:
1275:
1170:
1144:
1137:
1130:
1126:
1119:
1114:
1110:
1090:
1086:
1082:
1078:
1074:
1073:, each term
1068:
1057:
1052:
1028:
1026:
912:
904:
889:
880:
866:power series
859:
642:
627:
611:
583:
530:
517:
436:
420:
412:Pythagoreans
409:
327:
317:
300:Partial sums
287:
268:
168:
147:
145:
60:
50:
45:
37:
5204:Conditional
5192:Convergence
5183:Telescoping
5168:Alternating
5084:Pell number
4880:Terence Tao
4637:Smithsonian
4616:February 3,
4556:February 3,
4522:February 5,
4383:January 31,
4087:11336/46148
3837:January 26,
3696:January 30,
3645:, in which
3614:Smithsonian
3548:Ed Copeland
3526:produced a
3524:Numberphile
2543:G. H. Hardy
2214:The series
1929:, one gets
433:Summability
326:, etc. The
5448:Categories
5229:Convergent
5173:Convergent
4889:Luboš Motl
4845:(Week 213)
4841:(Week 147)
4837:(Week 126)
4665:August 31,
4642:2016-05-16
4490:, Oberon,
4059:Hardy 1949
3889:2007-03-22
3817:Hardy 1949
3665:References
3659:Mathologer
3478:Littlewood
2994:See also:
2448:asymptotic
568:Heuristics
40:-intercept
5260:Divergent
5178:Divergent
5040:Advanced
5016:Factorial
4964:Sequences
4860:John Baez
4854:John Baez
4210:CiteSeerX
3980:(1990) .
3919:(1990) .
3904:: 83–106.
3752:0908.0333
3285:⋯
3226:⋯
3167:−
3158:⋯
3096:⋯
3037:⋯
2945:−
2924:×
2911:−
2815:is the (2
2797:is the (2
2758:−
2705:∞
2690:∑
2686:−
2661:−
2617:∞
2602:∑
2375:∞
2360:∑
2315:∑
2104:−
2096:→
2077:⋯
2058:−
2033:−
2018:−
2010:→
1990:−
1984:η
1972:−
1966:ζ
1960:−
1897:η
1882:ζ
1871:−
1860:−
1815:η
1805:⋯
1794:−
1786:−
1778:−
1757:−
1749:−
1741:−
1720:−
1712:−
1704:−
1674:ζ
1661:−
1650:−
1634:⋯
1623:−
1615:×
1596:−
1588:×
1569:−
1561:×
1533:ζ
1525:−
1517:×
1506:⋯
1495:−
1479:−
1458:−
1442:−
1421:−
1405:−
1375:ζ
1250:−
1240:∞
1225:∑
1199:∞
1184:∑
1054:the step
962:⋯
953:−
941:−
926:−
888:but with
841:⋯
832:−
815:−
798:−
778:−
768:⋯
719:⋯
578:Ramanujan
427:term test
342:∑
241:−
235:⋯
78:∑
33:asymptote
5433:Category
5199:Absolute
4710:(1949).
4468:(2007),
4370:54514141
4303:L'Ouvert
4148:Zee 2003
4081:: 1–11,
3620:−
3616:magazine
3573:−
3501:−
3408:−
3343:, where
2858:−
2561:−
2551:Bromwich
2524:−
2489:−
2475:, where
2453:−
2273:−
2246:−
2177:−
1330:−
1149:−
1133:) > 1
1109:Plot of
1085:, where
1033:−
591:−
547:−
183:−
156:sequence
152:infinity
150:goes to
29:parabola
5209:Uniform
4783:Bibcode
4595:YouTube
4577:YouTube
4274:2319041
4232:2690371
4170:Bibcode
4110:Bibcode
3727:Bibcode
3657:on his
3635:
3588:
3566:
3554:
3528:YouTube
3516:
3423:
3402:History
3369:− 2)/24
3320:Physics
2873:
2845:
2833:
2576:
2539:
2504:
2468:
2440:bounded
2428:(0) = 1
2288:
2261:
2192:
2175:(−1) =
1935:(−1) =
1345:
1328:(−1) =
1164:
1147:(−1) =
1117:). For
1048:
909:
894:
886:
870:
606:
562:
513:
501:
491:
479:
471:
459:
198:
5161:Series
4968:series
4829:Blogs.
4755:
4727:
4696:
4494:
4368:
4272:
4230:
4212:
4180:
4023:
3996:
3992:–492.
3960:
3935:
3931:–476.
3799:
3774:
3649:means
3647:equals
3639:equals
3386:for a
2793:where
2442:, and
2436:smooth
2419:where
1320:(−1).
1316:to be
1122:> 1
535:below)
281:, and
164:series
162:, the
59:. The
35:; its
27:. The
5104:array
4984:Basic
4867:(PDF)
4852:– by
4798:, 1,
4773:arXiv
4377:(PDF)
4366:S2CID
4348:arXiv
4336:(PDF)
4310:(PDF)
4299:(PDF)
4270:JSTOR
4250:(PDF)
4228:JSTOR
3747:arXiv
3717:arXiv
879:(1 +
632:into
529:(see
457:, to
55:is a
5044:list
4966:and
4753:ISBN
4725:ISBN
4694:ISBN
4667:2023
4618:2014
4558:2014
4524:2014
4517:Plus
4492:ISBN
4425:2014
4385:2014
4318:2014
4281:2014
4178:ISBN
4134:See
4021:ISBN
3994:ISBN
3958:ISBN
3933:ISBN
3839:2014
3797:ISBN
3772:ISBN
3698:2014
3538:and
2885:) =
2819:)th
2805:and
2580:..."
2514:The
1939:(−1)
1927:= −1
1302:= −1
903:1 +
322:are
177:and
4887:by
4878:by
4825:",
4804:doi
4593:on
4575:on
4448:doi
4412:doi
4408:120
4358:doi
4262:doi
4220:doi
4118:doi
4106:190
4083:hdl
3990:490
3929:475
3598:In
3591:as
3357:nħω
3333:− 2
3324:In
2553:'s
2518:of
2450:to
2434:is
2089:lim
2003:lim
1308:by
1171:In
1140:= 1
541:or
499:to
5450::
4843:,
4839:,
4835:,
4781:.
4771:.
4635:.
4547:,
4532:^
4515:,
4442:,
4406:,
4402:,
4364:,
4356:,
4344:57
4342:,
4338:,
4301:,
4268:,
4258:81
4256:,
4252:,
4226:,
4218:,
4206:56
4204:,
4176:,
4164:,
4116:,
4104:,
4079:29
4077:,
4035:^
3902:17
3725:.
3672:^
3632:12
3585:12
3513:12
3420:12
3363:ħω
3176:0.
2953:12
2870:30
2856:=
2848:,
2831:=
2823::
2573:12
2536:12
2506:.
2501:12
2472:CN
2470:+
2465:12
2438:,
2302:.
2285:12
2263:.
2189:12
1949::
1931:−3
1342:12
1161:12
1097:.
1050:.
1045:12
762:12
603:12
559:12
515:.
473:.
429:.
292:,
285:.
277:,
249:12
195:12
5046:)
5042:(
4956:e
4949:t
4942:v
4869:.
4806::
4800:4
4789:.
4785::
4775::
4761:.
4733:.
4702:.
4645:.
4621:.
4597:.
4579:.
4561:.
4527:.
4501:.
4474:.
4455:.
4450::
4444:3
4428:.
4414::
4388:.
4360::
4350::
4321:.
4284:.
4264::
4235:.
4222::
4187:.
4172::
4138:.
4125:.
4120::
4112::
4090:.
4085::
4049:.
4029:.
4002:.
3967:.
3941:.
3892:.
3807:.
3805:.
3781:.
3755:.
3749::
3735:.
3733:.
3729::
3719::
3701:.
3629:/
3626:1
3622:+
3593:ζ
3582:/
3579:1
3575:+
3563:4
3560:/
3557:1
3510:/
3507:1
3503:+
3482:ζ
3417:/
3414:1
3410:+
3367:D
3365:(
3361:−
3353:n
3349:ω
3345:D
3331:D
3294:,
3291:0
3288:=
3282:+
3279:0
3276:+
3273:0
3270:+
3267:0
3264:+
3261:1
3235:,
3232:0
3229:=
3223:+
3220:1
3217:+
3214:1
3211:+
3208:1
3205:+
3202:0
3173:=
3170:x
3164:x
3161:=
3155:+
3152:1
3149:+
3146:1
3143:+
3140:1
3114:x
3111:=
3108:x
3105:+
3102:0
3099:=
3093:+
3090:3
3087:+
3084:2
3081:+
3078:1
3075:+
3072:0
3046:,
3043:x
3040:=
3034:+
3031:3
3028:+
3025:2
3022:+
3019:1
2979:f
2975:f
2958:.
2950:1
2942:=
2936:!
2933:2
2929:1
2919:6
2916:1
2908:=
2905:c
2892:f
2887:x
2883:x
2881:(
2879:f
2867:/
2864:1
2860:+
2854:4
2851:B
2842:6
2839:/
2836:1
2829:2
2826:B
2817:k
2812:k
2810:2
2807:B
2803:f
2799:k
2795:f
2778:,
2775:)
2772:0
2769:(
2764:)
2761:1
2755:k
2752:2
2749:(
2745:f
2738:!
2735:)
2732:k
2729:2
2726:(
2720:k
2717:2
2713:B
2700:1
2697:=
2694:k
2683:)
2680:0
2677:(
2674:f
2669:2
2666:1
2658:=
2655:c
2631:)
2628:k
2625:(
2622:f
2612:1
2609:=
2606:k
2590:f
2570:/
2567:1
2563:+
2533:/
2530:1
2526:+
2498:/
2495:1
2491:+
2485:f
2481:f
2477:C
2462:/
2459:1
2455:+
2432:f
2426:f
2421:f
2404:,
2400:)
2395:N
2392:n
2387:(
2383:f
2380:n
2370:0
2367:=
2364:n
2335:n
2330:N
2325:0
2322:=
2319:n
2282:/
2279:1
2275:+
2258:8
2255:/
2252:1
2248:+
2242:y
2194:.
2186:/
2183:1
2179:+
2173:ζ
2155:.
2150:4
2147:1
2142:=
2134:2
2130:)
2126:x
2123:+
2120:1
2117:(
2113:1
2100:1
2093:x
2085:=
2081:)
2074:+
2069:3
2065:x
2061:4
2053:2
2049:x
2045:3
2042:+
2039:x
2036:2
2030:1
2026:(
2014:1
2007:x
1999:=
1996:)
1993:1
1987:(
1981:=
1978:)
1975:1
1969:(
1963:3
1943:η
1937:η
1933:ζ
1925:s
1920:s
1906:)
1903:s
1900:(
1894:=
1891:)
1888:s
1885:(
1879:)
1874:s
1868:1
1864:2
1857:1
1854:(
1827:.
1824:)
1821:s
1818:(
1812:=
1802:+
1797:s
1790:6
1781:s
1774:5
1770:+
1760:s
1753:4
1744:s
1737:3
1733:+
1723:s
1716:2
1707:s
1700:1
1692:=
1683:)
1680:s
1677:(
1670:)
1664:s
1658:1
1654:2
1647:1
1643:(
1631:+
1626:s
1619:6
1612:2
1609:+
1599:s
1592:4
1585:2
1582:+
1572:s
1565:2
1558:2
1551:=
1542:)
1539:s
1536:(
1528:s
1521:2
1514:2
1503:+
1498:s
1491:6
1487:+
1482:s
1475:5
1471:+
1461:s
1454:4
1450:+
1445:s
1438:3
1434:+
1424:s
1417:2
1413:+
1408:s
1401:1
1393:=
1384:)
1381:s
1378:(
1358:s
1356:(
1354:η
1347:.
1339:/
1336:1
1332:+
1326:ζ
1318:ζ
1306:s
1300:s
1291:s
1287:s
1285:(
1283:ζ
1276:s
1258:.
1253:s
1246:n
1235:1
1232:=
1229:n
1204:n
1194:1
1191:=
1188:n
1167:.
1158:/
1155:1
1151:+
1145:ζ
1138:s
1131:s
1129:(
1127:ζ
1120:s
1115:s
1113:(
1111:ζ
1091:s
1087:s
1083:n
1079:n
1075:n
1058:c
1056:4
1042:/
1039:1
1035:+
1029:c
1012:.
1007:4
1004:1
999:=
991:2
987:)
983:1
980:+
977:1
974:(
970:1
965:=
959:+
956:4
950:3
947:+
944:2
938:1
935:=
932:c
929:3
913:x
905:x
900:/
897:1
890:x
883:)
881:x
876:/
873:1
838:+
835:6
829:5
826:+
818:4
812:3
809:+
801:2
795:1
787:=
784:c
781:4
775:c
765:+
759:+
751:8
748:+
740:4
732:=
729:c
726:4
716:+
713:6
710:+
707:5
704:+
696:4
693:+
690:3
687:+
679:2
676:+
673:1
665:=
662:c
643:c
600:/
597:1
593:+
556:/
553:1
549:+
510:4
507:/
504:1
488:2
485:/
482:1
468:2
465:/
462:1
395:.
390:2
386:)
383:1
380:+
377:n
374:(
371:n
365:=
362:k
357:n
352:1
349:=
346:k
328:n
254:,
246:1
238:=
232:+
229:4
226:+
223:3
220:+
217:2
214:+
211:1
192:/
189:1
185:+
148:n
131:,
126:2
122:)
119:1
116:+
113:n
110:(
107:n
101:=
98:k
93:n
88:1
85:=
82:k
61:n
38:y
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