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Volume 2, Issue 1

Letters and Notes

Divergent Series and Its Assigned
Value in a Hyperreal Context
Bas van der Reijden
DOI: 10.33014/issn.2640-5652.2.1.reijden.1

Abstract

This letter discusses the deep connection between the infi-
nite sum of natural numbers and the value ! 1

12 . Aside of
more widely known facts, we consider a nontrivial way in
which we show the veracity of this connection; more pre-
cisely this concerns the BGN method (Bartlett, Gaastra,
and Nemati, 2020) applied on the so-called damped oscil-
lated Abel summed variant of the series. Moreover, we have
found a generalization of this method which ‘correctly’ as-
signs finite values to other divergent series. We conclude
with some questions concerning whether and how we can
analytically relate our hyperreal terms to frame the method
in a more justifiable and applicable context.

Background

It is obvious that the sum of natural numbers 1+2+3+ · · ·
tends to infinity and can thus not be equal to ! 1

12 . There
does however exist some connection between this series and
value and it is highly probable that this connection is im-
plicitely used (i.e. ‘under the hood’) in e.g. physics (which
often turns out to be perfectly justifiable, as can be shown
by various experiments).

The first evidence of this connection is retrieved when one
considers the Riemann zeta function ! (s). It is known that
! (s) is equal to ! 1

12 when s = !1 and it is interesting that
one retrieves the sum of natural numbers when one ‘plugs
in’ s = !1 at the defining series of the Riemann zeta func-
tion

!"
k=1

1
k s . Plugging s = !1 in the above series is unfor-

tunately not justifiable (given that ! (s) is only equal to this
series when #(s) > 1) but it remains an interesting thing
to mention.

Another evidence of the connection can be revealed when

Figure 1: Smoothed partial sums
!N

k=1 k with a y-
intercept of ! 1

12 .

one considers the ‘smoothed version’ of the partial sums!N
k=1 k, see also Tao, 2010. It turns out that these smoothed

partial sums have the same behaviour as the regular partial
sums (i.e. they have the same asymptotic expansion) and
thus tend to infinity when N $ ". However, one probably
recognizes the constant value in its asymptotic expansion,
which is (according to Tao (2010)) given by CN2! 1

12+O( 1
N )

(with C some coe!cient of little importance in this case).
Moreover, it is given that ! 1

12 is attained when one looks
at the intersection with the y-axis in Figure 1.

Damped Oscillations

The first two evidences we mentioned are quite widely
known but a more unknown fact can be observed when
one considers a damped oscillating variant of the series
1 + 2 + 3 + · · · , namely

""

k=1

ke!k! cos(k" ). (1)



30 Letters and Notes

This variant was also discussed in a previous letter (Bartlett
and Khurshudyan, 2019). In this letter it was also men-
tioned that, in the context of hyperreal numbers by intro-
ducing # := " (i.e. by appying the BGN method on it), (1)
can be written in a closed-form expression (or at least as
an approximation of it). It remained however still unclear
which value/magnitude the infinitesemal quantity " must
be1 in order that the BGN method applied on (1) equals
the ‘appropriate’ value ! 1

12 ; only numerical evidence was
given. In particular, it was shown that if " = 1

" , the com-
puting software “Wolfram Mathematica” will include the
constant ! 1

12 in its BGN expression (which is similar to the
observation of the previous paragraph).

At the time that Bartlett and Khurshudyan (2019) was
written, it only seemed clear that there is a numerical evi-
dence that (1) equals ! 1

12 when we take " in a su!ciently
small interval. Recently, we have found that Sugiyama
(2014) (Section 2.3) provides a more theoretical deriva-
tion of this matter. Although the website and its choice
of words are somewhat vague and confusing, the derivation
seems correct. In this derivation there is being made use
of a so-called ‘damped oscillated Abel summation method’,
which is a kind of generalization of the more common Abel
summation method used to assign finite values to divergent
series. In this article, this method of ‘damped oscillated
Abel summation’ is consequently used on a larger class of
divergent series as well; furthermore it turns out that the
‘damping’ and ’vibrating’ constant should not be necessar-
ily equal to each other. We thus in fact have that (see also
Section 5.2 and Section 6.1 of Sugiyama (2014), we here
write " instead of x)

!"
k=1 k i can be transformed to (letting

i % 1 be an integer)

""

k=1

k ie!k! cot !
2i+2 cos(k" )

and
""

k=1

k ie!k
i+1
2 ! cos(k

i+1
2 " )

and consequently taking the limit " $ 0 yields the ‘appro-
priate’ assigned value; we also numerically verified this2.

It remains of course interesting how this damped oscillated
Abel summation method can be stated in our more ‘de-
tailed’ hyperreal context; i.e. in which we know the exact
values of " (possibly in terms of #) in order that the BGN
method assigns the ‘appropriate’ value to a divegent series.
Unless it is still untrivial which values " must have in order
that the mentioned method assigns this value, we can how-
ever say from Equation (5.57) in Sugiyama (2014) that in

1in relation to "
2by again letting ! be in a su!ciently small interval

general 1
! must be a lot smaller than # (this was also shown

by numerical experiments: if we set " = 0.01, # must be a
lot larger than 100).

Conclusion

In conclusion, we can thus say that the connection between
the often assigned value of a divergent series is hidden in
its asymptotic expansion. Furthermore, some slight varia-
tions (performed in the context of hyperreals) of the terms
in the divergent series will alterate its asymptotic expan-
sion in such a manner that that the BGN method assigns
the ‘desired value’ to it. As it is at this point still untrivial
when equality holds, and how in this case " and # thus
must be related, remains an interesting topic for further
research. To state this in a more general and mathemati-
cally way: Consider a divergent series with BGN expansion
A(" (#))#2 + C + O(1/#) (here A is a value dependent of "
which is in turn dependent of # and C is the ‘appropriate’
value we want to have), the question is now which varia-
tions (in terms of " (#)) we have to make in order to make
A(" (#))#2 equal to zero.

Bartlett, J, L Gaastra, and D Nemati (2020). “Hyperreal
Numbers for Infinite Divergent Series”. In: Communi-
cations of the Blyth Institute 2.1, pp. 7–16.

Bartlett, J and A Khurshudyan (2019). “Numberphile’s
Proof for the Sum 1+2+3+...” In: Communications of
the Blyth Institute 1.1, pp. 54–55.

Sugiyama, K (2014). New proof that the sum of natural
numbers is -1/12 of the zeta function. url: https :
/ / xseek - qm . net / Regularization _ e . htm #
_Toc524947400 (visited on 11/16/2019).

Tao, T (2010). The Euler-Maclaurin formula, Bernoulli
numbers, the zeta function, and real-variable analytic
continuation. url: https://terrytao.wordpress.
com/2010/04/10/the-euler-maclaurin-formula-
bernoulli - numbers - the - zeta - function - and -
real-variable-analytic-continuation/ (visited on
01/07/2019).


