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# Hilbert's Grand Hotel with a series twist

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## Abstract

This paper presents a new twist on a familiar paradox, linking seemingly disparate ideas under one roof. Hilbert's Grand Hotel, a paradox which addresses infinite set comparisons is adapted and extended to incorporate ideas from calculus – namely infinite series. We present and resolve several variations, and invite the reader to explore his or her own variations.
International Journal of Mathematical Education in Science and Technology, 2014
http://dx.doi.org/10.1080/0020739X.2013.872307
CLASSROOM NOTE
Hilbert’s Grand Hotel with a series twist
Chanakya Wijeratnea,AmiMamolo
band Rina Zazkisa
aFac ul ty o f Ed uc at io n, Si mo n Fra se r Un iv er si ty, 8 88 8 Un iv er si ty D ri ve , Bu rn aby, BC V5A 1S6,
Canada; bFac ul ty o f Ed uc at io n, Un iv er si ty o f On ta ri o In st it ut e of Te chnology, 11 Simcoe Stree t
North, P.O. Box 385, Oshawa, ON L1H 7L7, Canada
This paper presents a new twist on a familiar paradox, linking seemingly disparate
ideas under one roof. Hilbert’s Grand Hotel, a paradox which addresses inﬁnite set
comparisons is adapted and extended to incorporate ideas from calculus – namely
inﬁnite series. We present and resolve several variations, and invite the reader to explore
his or her own variations.
Keywords: inﬁnity; limits and series; Hilbert’s Grand Hotel; paradoxes; calculus
1. Introduction
In undergraduate mathematics, inﬁnity appears in two seemingly unrelated places: in cal-
culus when considering limits of functions and series, and in set theory when considering
cardinality. In this paper, we present several variations of the classic paradox, Hilbert’s
Grand Hotel, which were designed in order to develop ideas of convergence and divergence
of inﬁnite series – speciﬁcally, we aim to connect under the same roof the two afore-
mentioned approaches to inﬁnity. The use of paradoxes, such as Zeno of Elea’s famous
Dichotomy Paradox – that which is in motion must arrive at the half-way point before
arriving at the destination – is discussed in calculus texts and lessons to introduce the
idea of geometric series (e.g. [1]). We extend this approach as informed by the ﬁndings of
to elicit cognitive dissonance and provoke change in learners’ understanding. In this paper,
we present novel exercises for connecting and developing ideas regarding inﬁnite series
suitable for undergraduate students and recreational mathematicians.
Difﬁculties with inﬁnite series are commonplace in university mathematics courses,
and can be related to different aspects of the topic, including its close connection to inﬁnite
as functions. Ervynck,[5]reectingonhismanyyearsofteachingexperience,suggested
this difﬁculty was because ‘students’ fundamental concept of a function is weak’ (p.93).
He further suggests that their limited interpretation of this abstract concept is unlikely to
change if it has proven to help them ‘be successful in tests and “ﬁnals” [as] this is the
unique objective of a large majority of them [students]’ (p.95). Beaver [6]arguedthat
problems involving sequences and series carry different conceptual challenges than many
other problems in calculus as their solutions involve arguments which require more logical
supporting work. Such logical reasoning requires training, ‘like weight lifting to an athlete,
Corresponding author. Email: ami.mamolo@uoit.ca
C
"2014 Taylor & Francis
2Classroom Note
like practicing scales and vocabulary to a musician, like reading to a second grader’ ([6],
p.529). As such, the mathematics instructor is motivated to provide students with a variety
of opportunities to develop and reﬁne their reasoning with sequences and series as the
underlying logical structure has wide-reaching applicability in (and beyond) mathematics
courses. Martinez-Planell, Gonzalez, DiCristina, and Acevedo [7]foundthatconstructing
the notion of an inﬁnite series as a sequence of partial sums was also quite challenging,
even for graduate students in mathematics. They recommended that students be faced with
more situations that make use of the sequence of partial sums than are typically found in
undergraduate studies in order to develop a more robust understanding of the concepts. They
also suggested that valuable learning could occur if students are faced with problematic
situations, designed to produce cognitive conﬂict, after being introduced to the notion of
an inﬁnite series and its sum. Similarly, Lindaman and Gay [8]usedtelescopingseriesin
their study on improving the instruction of inﬁnite series, because it is one type of series
where the deﬁnition of series convergence as a limit of partial sums is used directly to show
convergence.
The variants of Hilbert’s Grand Hotel paradox that we develop below, some of which
involve telescoping series, can serve as a means to foster robust understanding by provoking
cognitive conﬂict, as an unusual review of convergence and divergence of inﬁnite series, or
alternatively as an introduction to these ideas.
2. Hilbert’s Grand Hotel and some familiar variants
Let us start by reminding the reader of the classical paradox and its resolution. Hilbert’s
Grand Hotel has inﬁnitely many rooms and they are alloccupied. One evening, an extremely
important guest arrives – and we let students worry whether this is the Queen of England
or B.B. King – and the hotel manager has to ﬁnd accommodation for this new guest. Of
course, no evicting guests or room sharing is allowed.
The standard mathematical solution vacates room number 1 for the celebrity, where
the previous occupant of room number 1 is relocated to room number 2, and all the other
guests are relocated to the room number 1 greater than their original room. More formally,
the solution makes use of the bijection f(n) =n+1torelocatealloftheguests.Research
has shown that undergraduate students struggle with this solution (e.g. [9]), but that the
paradox invites debate and discussion that fosters an understanding of the formal resolution.
Sierpinska,[10]inherdetailedstudyofhumanitiesstudents’epistemologicalobstacles
related to inﬁnity in calculus, noted the importance of such experiences, suggesting that ‘to
master the art of rational discussion, of reasonable argumentation, of conducting fruitful
debates, is something useful not only to future mathematicians but also, and perhaps even
more, to future humanists’ (p.396). In our teaching experiences, such debate and discussion
occur in students’ resolutions of well-known extension to Hilbert’s Grand Hotel. One such
popular extension deals with a number (k)ofcelebritiesthatarrive,resultinginhosting
them in rooms numbered 1 to kand in relocation of other guests (via the bijection f(n) =
n+k). However, in another extension an inﬁnite bus arrives carrying an inﬁnite number
of passengers who need accommodation. One possible resolution uses the bijection f(n)=
2n,whereineachguestisrelocatedtotheroomtwicethenumberoftheiroriginalroom,
leaving inﬁnitely many odd-numbered rooms to accommodate the passengers in the bus.
These debates of our students revolve around a seeming anomaly of inﬁnite sets. Namely,
each of these variants exploits the fact that a set with inﬁnitely many elements can be put into
one-to-one correspondence with one of its proper subsets. It is this fundamental property of
inﬁnite sets that guarantees that every original and new guest will have a room in the hotel
International Journal of Mathematical Education in Science and Technology 3
following the relocation. In lay terms, this means that both inﬁnite sets, that of original
guests and the set of original +new guests have the same ‘size’ (formally, the two sets
have the same cardinality), and as such allows the accommodation of as many as countably
inﬁnite new guests without evicting any of the original patrons. The consistent use of one-
to-one correspondence (or bijection) in comparing inﬁnite sets is a cornerstone to Cantor’s
Theory of Transﬁnite Numbers,[11] as it resolves the discrepancies when comparing sets
by strategies such as the ‘inclusion’ method (where a set which can be seen as ‘included’ in
another must be smaller in ‘size’). Part of students’ struggle with Hilbert’s Grand Hotel, as
well as with other questions that rely on the comparison of inﬁnite sets, is attributed to the
conﬂicting results obtained when comparing sets by the inclusion method versus through
one-to-one correspondence. Tsamir [12] pointed out that even sophisticated mathematics
students who had studied set theory ‘still failed to grasp one of its key aspects, that is,
that the use of more than one ...criteria for comparing inﬁnite sets will eventually lead to
contradiction’ (p.90). As such, these questions and paradoxes can offer powerful learning
experiences, both in terms of concepts of inﬁnity and sets, and in terms of important aspects
of mathematical reasoning.
Further, Hilbert’s Grand Hotel and its variants have engaged learners of both sophisti-
cated and novice mathematical backgrounds, encouraging reﬂection on prior knowledge,
stimulating debate, and uncovering tacit ideas that inﬂuence interpretations and resolu-
resolve the paradox is a powerful motivator for change of knowledge’ (p.284) and may
provoke ‘a transition to the stage of relational [deep conceptual] understanding’ (p.284).
In our experiences, the attempts to resolve these paradoxes have invited students to make
conjectures, challenge and refute ideas, seek generalizations, and take ownership for ideas
presented – activities that are in line with academic practices of mathematicians.[13]We
view the negotiation of paradoxes as powerful means for students to engage with mathe-
matics in sophisticated ways and propose the following paradox variants to encourage such
mathematical activities around the notions of convergence and divergence of inﬁnite series.
3. Introducing room fees
In this section we develop several variations to Hilbert’s Grand Hotel that parallel the
aforementioned formulations. To connect the above ideas with inﬁnite series, we introduce
anoveltwist.SupposethattheGrandHotelunderwentrenovationssuchthatoccupantsof
different rooms were charged different amounts for their accommodations. Room number 1,
the most lavish and fabulous of rooms, cost 1 Goldcrown. Room number 2 (which is still
pretty fabulous, but lacks a Jacuzzi bathtub and pay-per-view) costs 1/2 Goldcrown, room
number 3 costs 1/3 Goldcrown, and room number ncosts 1/nGoldcrown, for every nN.
When all the rooms are occupied, what is the revenue? Here we have a divergent series,
and this question may be used to either introduce the notion or as reminder of a previously
known fact:
The series
!
k=1
1
kis divergent (so the revenue must be inﬁnite).
4. Celebrities arrive: 1, 2, ...,korinnitelymany
However, as in the classical story, an important guest arrives and the hotel manager decides
to give him room number 1. Therefore, the guests will relocate to the next numbered room,
as before. Now, since room number nwas more expensive than room number (n+1) a
4Classroom Note
new question arises: What is the expected refund that the manager will have to pay all the
guests who have to move to cheaper rooms?
In this question, occupants have to be refunded the difference in cost between the
original room and the one newly assigned. At ﬁrst glance, the intuition might be that the
refund is rather large, maybe even close to the initial revenue. In our teaching experience,
most students initially believe the refund to be inﬁnite, a common argument being that:
‘There is no way to calculate the total refund as the number of guests is inﬁnite. If the
manager has to pay each guest the amount will grow inﬁnitely high ...Paying an inﬁnite
number of people will cost an inﬁnite amount of money.’ In fact, however, the total refund
is 1 Goldcrown as "n
l=1(1
l1
l+1)=11
n+1and limn→∞(1 1
n+1)=1.
Similar to the variation on the classical problem, we can ask now, what if two
celebrities arrive? Or what if kcelebrities arrive? Addressing the latter, we know that as
before, the occupant of room number nrelocates to room number n+k. As such, his
refund must be the difference between 1
nand 1
n+k.Inthiscase,thetotalvalueof
the refund is 1 +1
2+1
3+···+ 1
kGoldcrowns, since the nth partial sum of
"
n=1(1
n1
n+k)isatelescopingsum:"n
l=1(1
l1
l+k)=(1 1
1+k)+(1
21
2+k)+···+
(1
k1
k+k)+(1
k+11
(k+1)+k)+···+(1
n11
(n1)+k)+(1
n1
n+k)=1+1
2+1
3+···+
1
k(1
n+1+1
n+2+1
n+3+···+ 1
n+k)fornk, and limn→∞ (1 +1
2+1
3+···+ 1
k
(1
n+1+1
n+2+1
n+3+···+ 1
n+k)) =1+1
2+1
3+···+ 1
k.
However, all the previous calculations – as simple as they are – are unnecessary if we
adopt the following point of view: Before the celebrity arrived, every room was paid for.
After the Queen is settled in her room and refunds have been issued, still all the rooms
are paid for, other than room number 1. Therefore, the refund (or revenue lost) must be 1
Goldcrown – the cost of accommodating our new important guest for free. Such a view may
be generalized to make sense of the calculations for the case of kimportant new guests.
This point of view is helpful in provoking students to look beyond their initial reactions
and to stimulate debate about the approach, calculations, and resolutions.
Our next step is to accommodate inﬁnitely many guests. If we let them – as in the
classical case – move to odd-numbered rooms – what is the refund?
Not surprisingly, the refund is now inﬁnite, as
!
n=1#1
n1
2n\$=
!
n=1
1
2n=.
But this begs the question: must a refund to inﬁnitely many guests be inﬁnite in amount?
Now the above refund can be written as "
n=1
1
n"
n=1
1
2n,whichinformallycanbe
thought of as ‘inﬁnity minus inﬁnity’. ‘Inﬁnity minus inﬁnity’ is of course undeﬁned. In
the case of accommodating kcelebrities the refund is "
n=1
1
n"
n=1
1
n+k, which is also
‘inﬁnity minus inﬁnity’, and we saw earlier that this is equal to the sum 1 +1
2+1
3+···+ 1
k,
which is a ﬁnite amount. Having seen this previous result, it is natural for a student to be
unsatisﬁed with having to give an inﬁnite refund, even when accommodating inﬁnitely many
guests. The instinct seems to be that the hotel manager would want to minimize revenue
lost. We suggest this may be an attempt to extrapolate ﬁnite experiences to the inﬁnite
case, and note that a fruitful debate can arise if the idea is planted that in accommodating
inﬁnitely many new guests the refund could be either inﬁnitely many Goldcrowns, or
it could be a ﬁnite amount of Goldcrowns....Dependingofcourse,onhowcleverour
manager is!
International Journal of Mathematical Education in Science and Technology 5
Can our clever manager ﬁnd a way to accommodate an inﬁnite (but countable!) number of
new guests, such that the refund for original guests is ﬁnite?
We pre s ent b e low o ur own del ibe r ati o ns on t h is qu e sti on as well as s ome i d eas f o r
classroom implementation. When put to students, this problem invites them to conjecture
whether different solutions exist, and which solutions students prefer and why so. For
example, rather than using the familiar bijection f(n) =2n,studentsmightaskwhatifwe
relocated guests by using f(n) =3n,orf(n) =10n?Inthesecases,themathematicsunfolds
similarly.
The refund would be "
n=1(1
n1
kn )="
n=1
k1
kn =if the bijection f(n)=kn is used.
However, our manager is equally unhappy with this alternative, and it soon becomes clear
that the challenge is to create a method of accommodating new guests so that the series of
refunded Goldcrowns is convergent.
In our experience with these paradoxes, the exploration for an appropriate convergent
series can sometimes overshadow the original question. For instance, drawing on familiar
convergent series, the suggestion that if the new guests were sent to rooms numbered with
powers of 2, then the refund would be: "
n=1
1
2n=1, seems at ﬁrst to work well. Similarly,
if we let the new guests occupy rooms numbered with triangular numbers, this yields a
ﬁnite refund:
!
n=1
2
n(n+1)
!
n=1
2
n2=π2
3.
While these suggestions indeed exemplify convergent series and could be familiar
to students, they are not in accord with the storyline. That is to say, for the convergent
series offered, the assumption must be that occupants were asked to leave their rooms
to accommodate the new guests – which explains why each occupant receives a refund
corresponding to his room number.
However, in our previous solutions, the occupants were not asked to leave, but rather
were relocated to a new (less fabulous) room. Thus the refund would be the difference
between the new room and the original room (rather than the cost of the original room).
Consider for example reallocating guests to rooms with only square numbers. In this case,
the refund would be "
n=1
1
n"
n=1
1
n2.However,since"
n=1
1
n2=π2
6the refund has to
be inﬁnite and we can formally see this from the fact that "
n=1(1
n1
n2)="
n=1(n1
n2)
and by comparing the latter series with the harmonic series. Of course, if the prices of the
rooms to which guests were relocated add up to a ﬁnite number, then the refund will be
inﬁnite. As such, a different look at convergence and divergence of series is needed.
The challenge remains: Is it possible to relocate guests to accommodate inﬁnitely many
newcomers such that every guest has his own room and the refund to the original guests is
ﬁnite?
6. One way to keep the manager happy
In what follows we suggest a solution, and we invite readers (and their students) to ﬁnd
additional and alternative ones. The idea behind this solution is partitioning N,thesetof
natural numbers, into sets Akfor k=1, 2, 3, ... and relocating the occupant of room
number nin Akto room number n+k. However, a judicious partitioning of Nis required.
6Classroom Note
Let Ak={n|(k1)2+1nk2}for =1,2,3,....Itiseasytoseethatthesets
Akare disjoint and %
k=1Ak=N.Letf:NNbe deﬁned by f(n)=n+kfor nAk
for k=1,2,3,....Itiseasytoseethatthisfunctionfis well deﬁned and one-to-one.
Now, f(1)=2, f(n)=n+2for2 n4, f(n)=n+3for5n9, ....Therefore
f(1)=2, f(2)=4, f(3)=5, f(4)=6, f(5)=8, f(6)=9, ....Notethattherange
of fis {2,4,5,6,8,9,10,11,12,14 ...
}(the occupied room). Therefore, asking the person
who is in room number nto move to room number f(n)we can accommodate inﬁnitely
many new guests (without eviction) because after this new room arrangement, rooms with
numbers in the set {1,3,7,13,21,...
}will be vacant. Now the refund would be
!
k=1
k2
!
n=(k1)2+1#1
n1
n+k\$=
!
k=1
k2
!
n=(k1)2+1
k
n(n+k).
This refund is ﬁnite as the series is convergent. To see this, consider the following
calculations:
!
k=1
k2
!
n=(k1)2+1
k
n(n+k)
!
k=1
k2
!
n=(k1)2+1
k
((k1)2+1) &(k1)2+1+k'
=
!
k=1
(2k1)k
((k1)2+1) &(k1)2+1+k'.
Let us now compare the latter series with "
k=1
1
k2using the limit comparison test:
we will compare the series "
k=1akand "
k=1bk,where ak=(2k1)k
((k1)2+1)((k1)2+1+k)and
bk=1
k2.Now,
lim
k→∞
ak
bk
=lim
k→∞
(2k1)k
((k1)2+1)((k1)2+1+k)k2
=lim
k→∞
(2 1
k)
((1 1
k)2+1
k2)((1 1
k)2+1
k2+1
k)=2.
Therefore, since 2 <and "
k=1bk="
k=1
1
k2is convergent, then "
k=1ak=
"
k=1
(2k1)k
((k1)2+1)((k1)2+1+k)is convergent. Hence "
k=1"k2
n=(k1)2+1
k
n(n+k)is also conver-
gent by the comparison test because we saw before that
!
k=1
k2
!
n=(k1)2+1
k
n(n+k)
!
k=1
(2k1)k
((k1)2+1)((k1)2+1+k).
7. Concluding remarks: Hilbert, Cantor, and Newton under one roof
The importance of series and sequences in and beyond calculus stems from Newton’s early
description of functions as sums of inﬁnite series.[14]Studentdifcultiesunderstanding
inﬁnite series are well documented (e.g. [7]), and our paper offers instructor problems
which can be used to uncover, explore, or address such difﬁculties. The paradoxical nature
International Journal of Mathematical Education in Science and Technology 7
of inﬁnite series – particularly ones for which a ﬁnite sum exists – are encountered early
in university students’ experiences with calculus, and can, in our perspective, be used to
advantage. The counterintuitive aspect of an endless calculation summing to a ﬁnite number
is the basis for some of the earliest recorded paradoxes. Zeno’s paradoxes highlighted the
inherent anomalies of the inﬁnite and had such a profound impact on mathematics and
mathematical thought that Russell attributed to them ‘the foundation of a mathematical
renaissance’ ([15], p.347).
The intrinsic appeal of addressing and resolving paradoxes can be a powerful motivator,
and one which can challenge and extend student thinking (e.g. [9]). The paradoxes developed
in this paper make use of a familiar context – hotel accommodations and relocations,
purchases and refunds – to frame notions of convergence and divergence as they apply to
inﬁnite series and sets. An important contribution of our ‘twist’ is in connecting seemingly
disparate occurrences of inﬁnity in undergraduate mathematics under the same roof. The
scenarios we developed offer interesting problems for exploration, and can be further
tweaked, depending on the sophistication and willingness of the student, to create more
or less challenging interpretations and resolutions. Our problems contribute a means of
stimulating mathematical debate, fostering a culture of conjecture and justiﬁcation, or
might simply be a fun mental playground for the curious type.
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