v1i1 54 Letters and Notes Laplace, Pierre-Simon (2012). Pierre-Simon Laplace Philo- sophical Essay on Probabilities: Translated from the fifth French edition of 1825 With Notes by the Trans- lator. Vol. 13. Springer Science & Business Media. Levin, Leonid A (1984). “Randomness conservation inequal- ities; information and independence in mathematical theories”. In: Information and Control 61.1, pp. 15–37. Menabrea, Luigi Federico and Ada Lovelace (1842). Sketch of the analytical engine invented by Charles Babbage. Romer, Paul M (1992). “Two strategies for economic de- velopment: using ideas and producing ideas”. In: The World Bank Economic Review 6.suppl_1, pp. 63–91. Solow, Robert M (1957). “Technical change and the aggre- gate production function”. In: The review of Economics and Statistics 39.3, pp. 312–320. Suárez, Fernando F and James M Utterback (1995). “Dom- inant designs and the survival of firms”. In: Strategic management journal 16.6, pp. 415–430. Turing, Alan M (2009). “Computing machinery and intelli- gence”. In: Parsing the Turing Test. Springer, pp. 23– 65. Numberphile’s Proof for the Sum 1 + 2 + 3 + . . . Jonathan Bartlett and Asatur Khurshudyan DOI: 10.33014/issn.2640-5652.1.1.bartlett.3 In 2014, YouTube math vlogger Numberphile upset the am- ateur math world by declaring that the sum of all the nat- ural numbers (i.e., the positive integers, the infinite series 1 + 2 + 3 + . . .) is − 1 12 (Haran and Padilla, 2014; Haran, 2015). While this is indeed the result of the Riemann Zeta function applied to −1, we will show here that it is not the sum of 1 + 2 + 3 + . . .. The standard summation which the Riemann Zeta function is based on is simple: ζ (x) = ∞! n=1 1 nx . (1) For x > 1, (1) is well defined and makes a convergent series. For x ≤ 1, (1) no longer converges. For x = −1, (1) is equivalent to the series under consideration, 1 + 2 + 3 + . . .. The question is whether or not ζ (−1) is still equivalent to the series implied by (1). If it is, then 1+2+3+ . . . is indeed equal to − 1 12 . According to the video, which uses a proof based on the one originally given by Ramanujan, the proof that 1+2+3+. . . = − 1 12 can be shown by beginning as follows. First, start with the following series: S1 = 1 − 1 + 1 − 1 . . . (2) S2 = 1 − 2 + 3 − 4 . . . (3) S3 = 1 + 2 + 3 + 4 . . . (4) S1 has the well-known value of 1 2 and S2 has the well-known value of 1 4 . He then subtracts S3 − S2. Doing this yields the series 0 + 4 + 0 + 8 . . .. The error comes next. This is claimed to be equivalent to the series 4+ 8+ 12 . . ., which would be 4S3. This gives the equation S3 − S2 = 4S3. Because S2 = 1 4 , this can be then solved. S3 − 1 4 = 4S3 (5) 3S3 = − 1 4 (6) S3 = − 1 12 (7) As suggested the problem comes with stating that 0 + 4 + 0 + 8 + 0 + 12 . . . = 4 + 8 + 12 . . . . Volume 1, Issue 1 Numberphile’s Proof for the Sum 1 + 2 + 3 + . . . 55 Bartlett, Gaastra, and Nemati (2018) developed a tech- nique, which we can term the BGN technique, that assigns hyperreal values to divergent sums. Applying the BGN technique shows that, even though it may seem counter- intuitive, adding zeroes in the middle of an infinite sum changes the value of the sum, therefore invalidating the proof. According to the method given in Bartlett, Gaastra, and Nemati (2018), the sum of 1 + 2 + 3 . . . is the hyperreal value ω2 2 + ω 2 . The sum of 0+4+0+8+0 . . . is the hyperreal value ω2 2 + ω 2 − 1 4 , which, in fact, is the result of S3− 1 4 . Note that both of these series are essentially the same value, as the lower-orders of infinity are essentially noise compared with the highest order term, which is ω2 2 . The reason why ζ (−1) = − 1 12 while the series (1) doesn’t is that ζ (−1) is evaluated using the Zeta function’s analytic continuation (a modification of a function that expands its domain), not the series given in (1). The analytic contin- uation of ζ (the expanded expression that actually is valid for −1) is, according to Lavrik (2011), π−x/2Γ " x 2 # ζ (x) = 1 x(x − 1) + $ ∞ 1 % x−(1−x/2) + x−(1−(1−s)/2) & θ(x) dx, (8) where Γ is the Euler Gamma Function, and θ(x) is'∞ n=1 e−πn 2x . This is no longer identical to the original ex- pression given in (1). However, the question still remains why physicists can use − 1 12 as a stand-in for the sum of all natural numbers. As Haran and Padilla (2014) point out, in several aspects of physics, such as for the Casimir effect, when physicists need a sum of all natural numbers, the Zeta function can act as a stand-in and yield valid results. While no conclusive reason for this has been established, Vandegrift (2014) offers a numerical evaluation of a series that is very similar to the series 1+2+3 . . ., but is offset by a tiny complex component. Vandegrift has suggested the possibility that lim ϵ→0 ∞! n=1 ne−ϵn cos(ϵn) = − 1 12 . (9) This sum would be nearly identical to 1 + 2 + 3 . . . in its beginning, but begin to diverge for higher values of n. We investigated this possibility and found the following results: 1. For an infinitesimal ϵ (where ϵ = ω−1), the series actu- ally diverges. 2. Interestingly, in the evaluation of the series expansion of (9), even though it diverges to infinity, there is a component of it that is − 1 12 .2 3. For a finite ϵ , a wide range of values will produce re- sults near − 1 12 , though we did not yet find a value that produces this value exactly. ϵ ranging from 1 2 to 1 3750 seemed to be fairly close, while values outside this range started to stray. Bartlett, J, L Gaastra, and D Nemati (2018). “Hyper- real Numbers for Infinite Divergent Series”. In: arXiv 1804.11342. Haran, B (2015). “This Blog Probably Won’t Help”. In: Brady Haran Blog. url: http : / / www . bradyharanblog.com/blog/2015/1/11/this-blog- probably-wont-help. Haran, B and T Padilla (2014). “Astounding: 1 + 2 + 3 + 4 + 5 + . . . = -1/12”. In: YouTube Numberphile Chan- nel. url: https://www.youtube.com/watch?v=w- I6XTVZXww. Lavrik, A F (2011). “Zeta Function”. In: Encyclopae- dia of Mathematics. Springer. url: https : / / www . encyclopediaofmath . org / index . php / Zeta - function. Vandegrift, G (2014). “MATLAB/Divergent Series In- vestigations”. In: Wikiversity. url: https : / / en . wikiversity.org/wiki/MATLAB/Divergent_series_ investigations. 2Using the BGN technique, the expansion of the sum was found to be ( sin(1) e − cos(1) 2e ) ω2 + ( cos(1) 2e ) ω − sin(1) 12e − 1 12 where ω is the hyperreal infinite unit. Notice the last part of this term is − 1 12 . It is unclear the connection between this value and the Zeta function. Nonetheless, it is interesting that − 1 12 appears there.