


































v2i1


Volume 2, Issue 1

Hyperreal Numbers for Infinite Divergent Series
Jonathan Bartlett, Logan Gaastra, and David Nemati

DOI: 10.33014/issn.2640-5652.2.1.bartlett-et-al.1

Abstract

Treating divergent series properly has been an ongoing is-
sue in mathematics. However, many of the problems in
divergent series stem from the fact that divergent series
were discovered prior to having a number system which
could handle them. The infinities that resulted from di-
vergent series led to contradictions within the real number
system, but these contradictions are largely alleviated with
the hyperreal number system. Hyperreal numbers provide
a framework for dealing with divergent series in a more
comprehensive and tractable way.

1 The Problem of Infinite Series

Historically, infinities have led to many problems in math-
ematics. Infinities, when not handled carefully, easily lead
to contradictions and indeterminacies. Therefore, caution
has always been urged when dealing with infinite series.

This is especially true with divergent infinite series. Con-
vergent infinite series generally behave unproblematically
similar to the value that they converge to. Given a series
that converges to 2 and another series that converges to 3
then the sum of the values of the series will be 5 and their
product will be 6. Therefore, the nature of these series can
be summarized into a single number.

With divergent series, this is not so straightforward. A
lack of agreement on the rules for handling infinities had
led to numerous problems with handling divergent series.
If a series diverges to infinity, is it greater than or equal to
some other series that diverges to infinity? Can the terms
of the series be rearranged? Can their spacing be modified?
Is 1 + 1 + 1 + . . . equivalent to 1 + 0 + 1 + 0 + 1 + 0 + . . .?

Lack of answers to questions like this have stifled work in
divergent series, and have caused many mathematicians to
think of divergent series as invalid entities to work with
rigorously.

2 Working with Infinities

Many paradoxes exist with infinities. For instance, are
there the same number of positive even integers as positive
integers? There are an infinity of them, but does that make
them the same? It seems pretty obvious that, on a number
line, positive integers occur twice as often. However, there
are an infinite amount of both.

Cantor’s solution to this problem is to separate out the final
quantity of a set (the cardinality) from the arrangment of
a set (its ordinality). The cardinal numbers do not behave
in any way similar to real numbers. The ordinals, on the
other hand, behave in many ways similar to real numbers.
However, Cantor’s own system for ordinal arithmetic is dif-
ficult to use, and doesn’t translate well between transfinite
and regular real arithmetic.

The hyperreal number line has many similarities to Can-
tor’s ordinals, operating essentially at the level of “ordinal”
in Cantor’s system. However, the hyperreal number line
o!ers a way to do arithmetic with infinities in a way that
very closely matches real arithmetic through the use of the
transfer principle (Henle and Kleinberg, 2003). The trans-
fer principle states that any first-order proposition that is
true for the reals is also true for the hyperreals. This means
that the standard arithmetic principles for dealing with real
numbers will apply to hyperreal numbers as well.

The hyperreal number line operates with an infinite unit, !,
that represents an order of infinity.1 The way it is usually
handled, ! isn’t a specific number in the typical sense, but
rather more of a benchmark of infinity.

Previous work has shown that hyperreal numbers could be
a potential solution to how values of divergent series can
be represented (Gaastra, 2016).2 The present paper will
build on this original idea and establish a system for using

1The choice of character/typography for the unit varies with the
author. For instance, Keisler uses H (Keisler, 2012). ! was chosen
because of its historical connection with ordinal-type infinities.

2Other work worth mentioning in this area are (Paterson, 2018a)
and (Paterson, 2018b). In the current work, we will use a notation
similar to (Keisler, 2012) to notate hyperreal values, and show how
infinite series can be simplified to them. Paterson did the opposite, by
notating hyperreal values with the infinite sum that represents them.



8 Hyperreal Numbers for Infinite Divergent Series

hyperreal numbers to assign values to infinite series.

3 Hyperreals and Partial Sums

The vast majority of issues with divergent series comes with
the transition from partial sums to infinite sums. As long as
a series remains a partial sum, arithmetic with the series is
unproblematic. Therefore, it would be beneficial to develop
a system which matched the partial sum behavior of finite
sums, but allowed the result to be generalized to infinity.

The value of a partial sum of an infinite sequence of a given
length is sensitive to the order of the terms in the infinite
sequence. Imagine summing the first n terms of an infinite
sequence. The result will not be the same with di!erent
orderings of the infinite sequence. For instance, if I did
a partial sum of the first n terms of an infinite sequence,
then reversed the infinite sequence, the partial sum of the
first n terms of the reversed sequence will not necessarily
match the original partial sum. However, within the first
n terms, rearrangements can occur without consequence.
If the extent of the partial summation is unknown, then
it is also unknown the extent to which numbers can be
reordered.

For the same reason, tacking on zeroes to the beginning of
the series can potentially change the partial sum. There-
fore, although adding zeroes to the beginning of a series has
the appearance of being a null operation, because doing so
modifies the value of finite partial sums, it can also lead to
long-term changes in behavior.

Additionally, changing the number of terms in a partial
sum alters the value. Adding together the first n numbers
of a sequence will often yield a di!erent value than adding
together the first m numbers of the sequence.

In short, partial sum behavior is well-behaved, well-
understood, and well-regulated. By understanding diver-
gent series in terms of partial sums extended into the hy-
perreals, we will be able to deal with them more rigorously
and uniformally.

To understand many of the rules that will be developed
for infinite series, imagine that the rules are being built
for merely doing partial sums to an unknown parameter
k, where k at least acts like a particular finite value, but
is larger than any particular list index referenced by any
finite manipulation of the series.

Some of these formulas will be further reducible due to the
nature of the hyperreals, as will be discussed in Section 7.

4 Pinning Down !

Since ! operates as a benchmark instead of a number, the
first task is to identify the benchmark to associate ! with.
This is actually to some extent an arbitrary decision. Any
infinitely large value could be used to establish a baseline
!.

However, the value that seems most natural for ! (espe-
cially for summation) is the size of the set of positive inte-
gers. Therefore, ! will be used to refer to the total quantity
of positive integers.3

! = |N| =
!

i!N
1 (1)

Because of this, the notation used will be more specific
when writing summations. Instead of summing to the am-
biguous infinity, ", a summation to the specific infinity of
all positive integers, !, will be used. Therefore, the series
1 + 2 + 3 + . . . will be written as

!!

i=1
i (2)

(1) will establish the starting benchmark for relationships
among the di!erent series.

5 The Standard Summation

Because partial sums are dependent on length and order,
it is important to establish an o"cial standardization of
summation. That is,

!"
i=1

will be di!erent from
!"
i=0

. Even

though it looks like series with these types of sums will
have an identical number of terms (after all they both have
infinite terms), using this methodology the latter one will
actually have more elements than the former.

This is due to the principle established in Section 3. If,
instead of ! being infinite, pretend that ! was just an or-
dinary finite integer parameter.

Examine the series
!!

i=1
1. (3)

3There are some objections to equating a hyperreal number to a
cardinal number such as |N |. The specific identity of ! with |N | is
for conceptual convenience. Alternatively, simply treating ! as an
arbitrary (but unchanging) benchmark of infinity leads to the same
results.



Volume 2, Issue 1

6. SIMPLE ARITHMETIC AND GEOMETRIC SERIES 9

If ! represented an integer (say, 5) instead of ", it would
be obvious that this sum represents a di!erent value from
the series

!!

i=0
1. (4)

Equation 3 would represent the value 5 while Equation 4
would represent the value 6. Therefore, it is clear that
having matching indices matters.

In fact, our ability to sum divergent series will sometimes
depend on having summations with equivalent numbers of
terms. Therefore, a “standard” starting point for summa-
tion will need to be established in order to ensure that like
entities are being compared and reasoned about. Since !
has been defined as being the size of the set of all positive
integers, it makes sense to start at 1. For the purposes of
this paper, the “standard” way of summing will be to start
with 1 and proceed to !.

6 Simple Arithmetic and
Geometric Series

6.1 Arithmetic Series

Arithmetic series take the form
n!

i=1
a + (i # 1)d. (5)

The sum of an arithmetic series, given a starting value a,
the number of elements n, and distance between elements
d, can be given by the formula

n!

i=1
a + (i # 1)d =

n
2

(2a + (n # 1)d) . (6)

To find the sum of an infinite arithmetic series, ! is used
for n, forming a hyperreal value. That reduces the formula
to

!!

i=1
a + (i # 1)d = !a +

!2d
2
# !d

2
. (7)

Therefore, to find the summation of the series 1+1+1+ . . .,
one must only substitute in the correct parameters. Since
the starting value is 1 and the distance between terms is 0,
this yields

!!

i=1
1 = ! · 1 + !

2 · 0
2
# ! · 0

2
(8)

= ! + 0 # 0 (9)
= !. (10)

It is intuitively obvious that since there are ! 1s added
together that the sum of them would add up to !, as would
be true for any finite value as well. This matches the value
given by equivalent considerations in (1).

The arithmetic series 1+ 2+ 3+ . . . can be calculated using
hyperreals as well.

!!

i=1
i = ! · 1 + !

2 · 1
2
# ! · 1

2
(11)

=
!2

2
+
!

2
. (12)

The next arithmetic series to examine is 1+3+5+ . . ., which
can be similarly calculated.

!!

i=1
(2i # 1) = ! · 1 + !

2 · 2
2
# ! · 2

2
(13)

= !2. (14)

Thus, the value of 1 + 3 + 5 + . . . is equal to (1 + 1 + 1 . . .)2.

Interestingly, as noted in Section 3, there is nothing intrin-
sically infinite about the behavior of ! in these series. For
instance, if ! was replaced with 5, the results would hold.
That is, (1 + 1 + 1 + 1 + 1)2 = (1 + 3 + 5 + 7 + 9) = 25.

Even though the sums are divergent, summing them has
a very well-defined behavior within the combined hyper-
real/partial sum methodology presented here.

6.2 Geometric Series

Geometric series take the form
n!

i=1
ar i#1, (15)

where n is the number of terms, a is the starting term, and
r is the common ratio.

A value for a geometric series can be given by the formula
n!

i=1
ar i#1 = a

1 # rn

1 # r
. (16)

Because an infinite series will have ! terms, n can be re-
placed with !.

Let us begin by looking at the series 1+ 2+ 4+ 8+ . . .. The
value of this series can be given by the formula

!!

i=1
2i#1 = 1 · 1 # 2!

1 # 2
(17)

= 2! # 1. (18)



10 Hyperreal Numbers for Infinite Divergent Series

Divergent geometric series will generally have the same
form.

Convergent series are also interesting. The series 1+ 1
2 +

1
4 +

. . . can be plugged into the formula to yield

!!

i=1

1
2

i#1
= 1 ·

1 # 1
2
!

1 # 1
2

(19)

= 2 # 2 ·
#1
2

$!
(20)

7 Generalizing to the Principal
Value

In most discussions of hyperreal numbers, the halo of a
number is considered the hyperreal values which are in-
finitely close to a standard real number. However, this def-
inition is too focused on real numbers.

We will consider the order of a hyperreal value to be its
largest exponent of !. This is the most significant term
of the hyperreal value. We will call this most significant
term the principal value of the hyperreal. The halo (also
known as a monad) of a hyperreal consists of all of the
hyperreals which have the same principal value.4 We will
use the $ operator to denote two hyperreals which share the
same principal value.5 Therefore, the halo of a hyperreal
number consists of all of those numbers which share the
same principal value.

Many people use “infinitely close” as a colloquialism to de-
scribe two hyperreals which share the same principal value.
However, technically it is not correct, since, when dealing

4Most texts on hyperreal numbers define the halo or monad of x to
be all of the values y for which x # y is infinitesimal (Loeb and Wol!,
2015, pg. 21) (Goldblatt, 1998, pg. 52). However, defined in such
a way, the infinitesimals !#1 and 2!#1 are within a monad. Using
principal values, !#1 and 2!#1 are in the same galaxy, but not the
same monad. You would have to have a term of lower-order infinity
to be within a monad, such as !#1 and !#1 + !#2. This seems to
be the essence of what the other texts are getting at, but, since most
mathematics focuses on the reals, their definitions were entirely based
on using reals as a starting point. Here, since we will have results in
the hyperreals, we need definitions that are equally useful when the
final result is a hyperreal number.

5In practice, $ can be replaced with =, as it denotes equality to the
extent normally practiced in mathematics. For instance, the di!eren-
tial d (xy) is often stated as being equal to x dy + y dx, but really it is
just the principal value. The actual value is x dy + y dx + dy dx. The
dy dx term is always discarded because it is infinitely less significant
than the other pieces. Even when discarding this term, the equality
sign is used. Therefore, while the present paper will be pedantic about
asserting exact equality or mere principal value, for most general pur-
poses equality can be asserted even when only stating the principal
value.

with infinities, two hyperreals which di!er by multiple in-
finities can be considered “infinitely close.” That is, !2+5!,
!2 # 12!, and !2 + 23 all share the same principal value,
!2. They are infinitely apart, yet, colloquially, they can
be considered “infinitely close” because their di!erences are
infinitely less significant than their similarities.

When dealing with hyperreals, the principal value is the
main one of concern. So, for instance, while 1 + 2 + 3 + . . .
is exactly described by !2

2 +
!
2 , its principal value is just

!2

2 . Therefore, the formula given in (7) can actually be
simplified to

!!

i=1
a + (i # 1)d $ !

2d
2

(21)

if d ! 0.6

Interestingly, we can see that, while the exact value of the
hyperreal associated with a series depends on the starting
point, the principal value depends only on the distance cho-
sen, provided that d ! 0.

Geometric series can use similar considerations. You may
have noticed that the hyperreal given for the series 1 + 1

2 +
1
4 + . . . in Section 6 is 2 # 2 ·

%
1
2

&!
. Typically, this series is

thought to converge to 2. In fact, its principal value is 2,
because

%
1
2

&!
is an infinitesimal.

The use of principal values allows for a great amount of
simplification for hyperreal values and formulas.

As an example, the ratio between two given arithmetic se-
ries can be solved for very simply.

S1 =

!!

i=1
a1 + (i # 1)d1 $

!2(d1)
2

S2 =

!!

i=1
a2 + (i # 1)d2 $

!2(d2)
2

S1

S2
$

!2 (d1)
2

!2 (d2)
2

=
d1

d2
(22)

In other words, the principal value of the ratio of two arith-
metic series is simply the ratio of the distances.

6When d = 0, then the !2 term goes to zero, and the series sim-
plifies to a ·! instead.



Volume 2, Issue 1

8. SERIES MANIPULATION RULES FOR FINITE SUBSETS 11

8 Series Manipulation Rules for
Finite Subsets

Many attempts to manipulate divergent series have resulted
in contradictions, to the extent that many suggest that it
is best to not attempt to do so. The reason for these con-
tradictions, however, lies in the treatment of the infinite
nature of the number of values.

In the real system, " is considered a boundless number.
That is, there is not "+1 that is distinct from ". Likewise,
"#1 is also infinity. Essentially, within the real numbers, "
is used largely like an ambiguous infinite value, essentially
saying that “the real numbers can’t handle this value.”

If, instead, the hyperreal numbers are used, then ! and
! + 1 are distinct quantities, despite the fact that they are
both infinite. The rules for manipulating series come from
these ideas. See Section 11 for a possible exception to these
rules.

8.1 Finite Term Addition

To begin with, it is possible to easily add a scalar value to
a series, provided that it is added to one of the particular
terms of the series. In other words, suppose the value A is
added to the series 1 + 2 + 3 + . . .. This can be written as

A +
!!

i=1
i (23)

or as
A + (1 + 2 + 3 + . . .). (24)

To integrate A into the series, A can be added to any distinct
position. The series could read as

(A + 1) + 2 + 3 + . . . (25)

or
1 + 2 + (A + 3) + . . . . (26)

All of these yield the same value for the final series, as long
as partial sums are taken starting after the index where A
is added.

Additionally, A can be spread across multiple finite terms.
For instance, half of A could be added to each of the first
two terms, yielding

A + (1 + 2 + 3 + . . .) =
'
1 +

A
2

(
+

'
2 +

A
2

(
+ 3 + . . . . (27)

In fact, there is no reason why the same amount would have
to be distributed to each position.

A + (1 + 2 + 3 + . . .) =
#
1 +

2
5

A
$
+

#
2 +

3
5

A
$
+ 3 + . . . (28)

8.2 Finite Term Insertion and Removal

Because this method of summation is based on partial sums,
it should be apparent that inserting and removing terms
will in fact alter the summation. For instance, let’s begin
with the arithmetic sum 1+1+1+ . . .. It may seem intuitive
that one should be able to freely add or remove a 1 from
this sum without a!ecting the sum. In this particular series,
the exact hyperreal value does change, but not the principal
value.

Again, remember that, as mentioned in Section 3, this con-
ception of summation will be based on partial sums. So, let
us begin by considering the partial sum

k!

i=1
1. (29)

If k is a finite number, then adding one to this sequence
will in fact alter its value. Additionally, removing a 1 from
this sequence will also alter its value. Therefore,

k!

i=1
1 ! 1 +

k!

i=1
1. (30)

Likewise,
k!

i=1
1 !

k!

i=0
1 !

k!

i=2
1. (31)

Because performing these operations will change the value
for any partial sum of k terms for a finite k, they will also
change the value for a hyperreal k such as !. However, for
these particular series, the principal value will be the same,
because ! $ ! + 1 $ ! # 1.

Additionally, a more surpising fact is that removing a term
from a sequence also changes its value if it does not also
change the number of terms being summed. Consider the
series

1 + 2 + 3 + . . . =
!!

i=1
i. (32)

This series is not equal to the series

1 +
!!

i=1
(i + 1). (33)



12 Hyperreal Numbers for Infinite Divergent Series

although it does have the same principal value in this case.
In other words,

(1 + 2 + 3 + . . .) ! 1 + (2 + 3 + 4 + . . .) (34)

but
(1 + 2 + 3 + . . .) $ 1 + (2 + 3 + 4 + . . .). (35)

The reason for this is readily apparent when considering
how these work in terms of partial sums. If the parameter
k was used instead of !, then it is apparent that the value
of (33) actually has an extra term compared to (32). That
is, it is obvious that

5!

i=1
i ! 1 +

5!

i=1
(i + 1). (36)

This can also be seen in the results of applying the arith-
metic series formula to the two series. For (1 + 2 + 3 + . . .)
the formula yields !2

2 +
!
2 . However, for (2+ 3+ 4+ . . .) the

formula yields !2

2 +
3
2!.

Now, terms can be removed without even a!ecting the ex-
act hyperreal value if they are replaced by zeroes in the
sequence, or if the sequence starting index is moved appro-
priately. In other words,

(1 + 2 + 3 + . . .) = 1 + (0 + 2 + 3 + . . .) = 1 +
!!

i=2
i. (37)

This can be easily proved using the principle derived in
Section 8.1. For instance, to move the 1 outside of the
series, 1 + #1 can be added to the series.

1 + #1 + (1 + 2 + 3 + . . .)
= 1 + ((1 + #1) + 2 + 3 + . . .)
= 1 + (0 + 2 + 3 + . . .) (38)

8.3 Finite Term Rearrangement

As can be deduced from Sections 8.1 and 8.2, any number of
finite terms in a series can be rearranged in position. That
is, for any given series member with a value of A, A # A
can be added to the series, applying the #A such that it
cancels out the value of the series member. After doing
this to several series members, the inverse operations can
then be applied to move these values to any finite position
in the series.

Doing this will preserve the partial summing behavior of
the series for all partial sums after the members which have
been manipulated.

9 More Advanced Series

While basic formulas for divergent series of arithmetic and
geometric series can be established using the standard for-
mulas, more advanced series require the use of discrete in-
tegral calculus7 to establish the formulas for such series.
Doing so leads to very interesting results.

9.1 Cesàro Sums and Oscillating Series

Oscillating series have an interesting history of treatment
within mathematics. The standard series to consider is
Grandi’s series: 1#1+1#1+ . . .. Or, written more formally,

"!

i=1
(#1)i+1. (39)

Partials sums for this series can be found by performing a
discrete integral.

n!

i=1
(#1)i+1 =

1
2

(#1)n+1 +
1
2
. (40)

What is particularly interesting about this formula is that
the Cesàro sum of the infinite series ( 1

2 ) is present in the
formula.

Now, consider the oscillating series #1 + 1 # 1 + . . .. This
series has the formula

"!

i=1
(#1)i . (41)

A discrete integral of the partial sums yields the formula

n!

i=1
(#1)i =

1
2

(#1)n # 1
2
. (42)

Note that in this as well, # 1
2 is the Cesáro summation of

the infinite series.

This leads to the conjecture that, in evaluating infinite se-
ries using integral formulas,

(#1)" = 0, (43)

at least for additive o!sets of !. For instance, in the case
of Grandi’s series, using the ! notation, the infinite series
would include (#1)!+1. The other series includes (#1)! .
According to the present conjecture, both of these simplify

7Also known as symbolic summation. See, for instance, Chapter 2
of Graham, Knuth, and Patashnik (1994).



Volume 2, Issue 1

9. MORE ADVANCED SERIES 13

to 0, at least for the purpose of creating formulas for infinite
series based on partial sums.

This can be understood probabilistically. Since we have no
information about what sign #1! will have, we can say that

#1! = ±1. (44)

Since both of these possibilities are equally probable, the
limit towards infinity resolves to their average, or zero.
Also, since we have no information about the sign of #1! ,
we have equally little information about the sign of #1!+1,
or any other variation on ! which is not biased towards
evenness (e.g., 2!).

The expression #1x has an oscillation pattern very similar
to sin(x). Since (Paterson, 2018a) showed that sin(!) = 0 in
the surreal numbers, it is possible that a similar proof may
be found for #1! = 0 along similar lines in the hyperreals.

9.2 Other Oscillatory Behavior

Because (a) discrete integration can be used to find formu-
las for series involving partial sums, and (b) the behavior
of (#1)" (for infinities without bias towards evenness) is
conjectured to be zero, the behavior of a wide variety of
oscillatory behaviors can be deduced.

Raising #1 to the ith power can produce all sorts of oscilla-
tory behavior. As has been seen with Grandi’s series, this
can produce a series of values that go back-and-forth across
a mean value (the mean value can be changed by adding,
and the back-and-forth can be changed by multiplying).

However, (#1)i can also be expanded to blank out members
of a series. For instance, to blank out every other member
of a series, the formula

((#1)i + 1)
2

(45)

can be used. This simplifies to 1 where i is even and 0 when
i is odd. Therefore, by multiplying a given formula by (45),
odd-indexed terms of the given formula will be zeroed out.

For instance, take the series 1 + 2 + 3 + . . .. This series can
be converted to the series 0+2+0+4+0+6+ . . . by applying
(45). This gives the series

!!

i=1
i ·

'
((#1)i + 1)

2

(
. (46)

The discrete integral yields
n!

i=1
i ·

'
((#1)i + 1)

2

(
=

1
8

%
2n2 + 2n(#1)n + 2n + (#1)n # 1

&

(47)
When n = ! the formula runs into a problem with sim-
plifying this through the conjecture (43) because it yields
an indeterminate form. The term 2n(#1)n becomes an in-
determinate form of the type ! · 0. This can be resolved,
however, through L’Hospital’s Rule.

lim
n%"

2n
(#1)#n

=
2

# ln(#1)(#1)#n
= # 2

ln(#1)
(#1)n . (48)

Now (43) can be applied without ambiguity, simplifying it
to zero.

Therefore, for n = !, (47) simplifies to
n!

i=1
i ·

'
((#1)i + 1)

2

(
=

1
8

%
2n2 + 2n # 1

&
. (49)

This means that the value of this sum in the hyperreals is
1
4!

2 + 1
4! # 1

8 $ 1
4!

2.

Interestingly, this is a di!erent result than for the simple
series 2+4+6+ . . .. Since 2+4+6+ . . . is a simple arithmetic
series, we can determine the hyperreal sum using (7).

!!

i=1
2 + (i # 1)2 = !2 + ! $ !2. (50)

This is a di!erent result than what was obtained for 0+ 2+
0 + 4 + 0 + 6 + . . ., which was 1

4!
2, indicating that the two

series have di!erent behaviors.

9.3 1 # 2 + 3 # 4 + . . .

Euler’s sum for the series 1#2+3#4+ . . . can be confirmed
using this method as well. This series can be given the
value

n!

i=1
i(#1)i#1 =

1
4

%
#2n(#1)n + (#1)n+1 + 1

&
. (51)

Using (43) and (48) this simplifies to 1
4 .

Interestingly, this is a series that is not changed even in its
exact hyperreal by prepending a zero to the function.

n!

i=1
(i # 1)(#1)i =

1
4

%
2n(#1)n + (#1)n+1 + 1

&
. (52)

Likewise, (43) allows this to reduce to 1
4 .



14 Hyperreal Numbers for Infinite Divergent Series

10 Whole Series Manipulation
Rules

In addition to manipulation of finite partial sums of a se-
ries, certain operations can (and can’t) be performed to the
series as a whole. In this section, some of these operations
will be considered.

10.1 Scalar Multiplication

Because of the distributivity of multiplication, multiplica-
tion of a series by a scalar value will distribute the scalar
multiplication to every term.

2(1 + 2 + 3 + . . .) = (2 · 1 + 2 · 2 + 2 · 3 + . . .). (53)

Or, written as a formula,

n
!!

i=1
f (i) =

!!

i=1
n f (i). (54)

10.2 Whole Series Addition

Adding two series together is equivalent to a term-by-term
addition of the series. Since the method presented here
is based on partial sums, term-by-term addition only works
when the lower and upper bounds of the terms are identical.

Therefore,

!
"

!!

i=1
f (i)#$ +

!
"

!!

i=1
g(i)#$ =

!!

i=1
f (i) + g(i). (55)

However,

!
"

!!

i=0
f (i)#$ +

!
"

!!

i=1
g(i)#$ !

!!

i=1
f (i) + g(i) (56)

because the limits of summation di!er. Again, to see why
this is the case, imagine replacing ! with a fixed scalar such
as 5. In (56), the left-hand addend would have a di!erent
number of terms than the right-hand addend.

10.3 Series Spacing

As noted in Section 8.2, adding or removing elements of a
series, even if they are zero, has an e!ect on the sum of
the resulting series. This e!ect can be calculated using the
considerations discussed in Section 9.

For instance, the series 1 + 1 + 1 + . . . can be spaced out by
adding in zeroes, to make 1 + 0 + 1 + 0 + . . .. A variation of
the oscillatory pattern in (45) can be used to give the series
the formula

n!

i=1

((#1)i+1 + 1)
2

. (57)

The discrete integral of this yields the formula
1
2

n +
1
4

(#1)n+1 +
1
4

(58)

Using conjecture (43) this reduces to the hyperreal value
1
2! +

1
4 $ 1

2!.

This is a slightly di!erent value (but with the same principal
value) than for the series 0 + 1 + 0 + 1 + . . .. This series can
be represented as

n!

i=1

((#1)i + 1)
2

=
1
2

n +
1
2

(#1)n # 1
4
. (59)

Using conjecture (43), the hyperreal value for this is 1
2! #1

4 $ 1
2!.

If (58) and (59) were added, it should be equivalent whether
they are added term-by-term (Section 10.2) or by summing
their relevant values.

Summing term-by-term it is apparent that

(1 + 0 + 1 + 0 + . . .)
+ (0 + 1 + 0 + 1 + . . .)
= (1 + 1 + 1 + 1 + . . .). (60)

The value of this series was deduced to be ! in (3). Like-
wise, if the values for each series are added the result is

#1
2
! +

1
4

$
+

#1
2
! # 1

4

$
= !. (61)

11 Ongoing and Future Work

11.1 Proving #1" = 0

The first obvious point of future work is the proof of the
conjecture in (43). Work on this proof is ongoing and is
promising.

11.2 Representing Infinitesimal Values

In general, the methods in this paper are about representing
infinite values using a series of finite terms. However, it



Volume 2, Issue 1

REFERENCES 15

may also be possible to write an infinitesimal value in a
similar way. While it is outside the scope of the present
paper, there is some evidence that, for instance, the series
1 + #1 + 0 + 0 + 0 + 0 + . . . (with the zero repeating forever)
represents an infinitesimal value.

This means that the rules established in Section 8 are lim-
ited to cases where the principal value is finite or infinite.

12 Conclusion

Here a method of summation was presented that uses the
structure of the hyperreal numbers to represent values for
divergent series. This methodology was shown to be sta-
ble across a variety of di!erent scenarios. One unproven,
but seemingly correct, conjecture was relied upon for this
formulation. Future work will focus on proving (43).

13 Acknowledgements

I wanted to take a moment to thank Stanley Schmidt. I was
thinking on this problem at the same time I was reading his
Life of Fred books to my children. The fundamental idea
for this method of summation came from thinking about
Gaastra’s original presentation (Gaastra, 2016) while read-
ing Life of Fred: Kidneys to my children, when Fred was
using the formula for arithmetic series (Schmidt, 2012).

Additionally, The Infinite by A. W. Moore provided some
help to the imagination in his discussion of the Löwenheim-
Skolem theorem. The basics of the discussion was to point
out that there was little in the theory of infinities that were
really unique to infinity. Even finite sets can look “infinite”
in some ways to other sets. The techniques and ideas ex-
plored in Section 3 were based largely o! of thinking about
infinities as much more tame and finite-like than is normally
considered.

Finally, I want to thank Jessica Hastings, whose interest
in the “Wheat and Chessboard” problem(Weisstein, 2018)
originally introduced me to the concepts in discrete calcu-
lus.

References

Gaastra, L (2016). “Omega: How Hilbert’s Infinite Hotel
Can Be Used to Evaluate Divergent Series”. In: GRCC
Student Mathematics Seminars. url: https://www.
youtube.com/watch?v=nlwh9oYiQYE.

Goldblatt, R (1998). Lectures on the Hyperreals: An Intro-
duction to Nonstandard Analysis. Springer.

Graham, R L, D E Knuth, and O Patashnik (1994). Con-
crete Mathematics. New York: Addison-Wesley.

Henle, J M and E M Kleinberg (2003). Infinitesimal Calcu-
lus. Dover Publications.

Keisler, J (2012). Elementary Calculus: An Infinitesimal
Approach. Second. Dover Books.

Loeb, P A and M P H Wol! (2015). Nonstandard Anal-
ysis for the Working Mathematician. second. London:
Springer.

Paterson, D A (2018a). “Banishing Divergence Part 1: Infi-
nite numbers as the limit of sequences of real numbers”.
In: arXiv 1108.5081v1. url: https://arxiv.org/abs/
1108.5081.

Paterson, D A (2018b). “Banishing Divergence Part 2:
Limits of Oscillatory Sequences and Applications”. In:
arXiv 1108.4952v1. url: https://arxiv.org/abs/
1108.4952.

Schmidt, S (2012). Life of Fred: Kidneys. Polka Dot Pub-
lishers.

Weisstein, E W (2018). “Wheat and Chessboard Problem”.
In: Mathworld—A Wolfram Web Resource.


