


































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.


