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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

b∗and 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

(Received 30 July 2013)

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

Movshovitz-Hadar and Hadass [2]whonotedparadoxesareaneffectivepedagogicaltool

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

sequences. For instance, Mamona,[3,4]foundstudentshaddifﬁcultythinkingofsequences

as functions. Ervynck,[5]reﬂectingonhismanyyearsofteachingexperience,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

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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

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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-

tions of the paradoxes. Movshovitz-Hadar and Hadass [2] observed that the ‘impulse to

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 n∈N.

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, ...,korinﬁnitelymany

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

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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

l−1

l+1)=1−1

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

n−1

n+k)isatelescopingsum:"n

l=1(1

l−1

l+k)=(1 −1

1+k)+(1

2−1

2+k)+···+

(1

k−1

k+k)+(1

k+1−1

(k+1)+k)+···+(1

n−1−1

(n−1)+k)+(1

n−1

n+k)=1+1

2+1

3+···+

1

k−(1

n+1+1

n+2+1

n+3+···+ 1

n+k)forn≥k, 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

n−1

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!

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International Journal of Mathematical Education in Science and Technology 5

5. The manager’s new task

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

n−1

kn )="∞

n=1

k−1

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

n−1

n2)="∞

n=1(n−1

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.

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6Classroom Note

Let Ak={n|(k−1)2+1≤n≤k2}for =1,2,3,....Itiseasytoseethatthesets

Akare disjoint and %∞

k=1Ak=N.Letf:N→Nbe deﬁned by f(n)=n+kfor n∈Ak

for k=1,2,3,....Itiseasytoseethatthisfunctionfis well deﬁned and one-to-one.

Now, f(1)=2, f(n)=n+2for2 ≤n≤4, f(n)=n+3for5≤n≤9, ....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=(k−1)2+1#1

n−1

n+k$=

∞

!

k=1

k2

!

n=(k−1)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=(k−1)2+1

k

n(n+k)≤

∞

!

k=1

k2

!

n=(k−1)2+1

k

((k−1)2+1) &(k−1)2+1+k'

=

∞

!

k=1

(2k−1)k

((k−1)2+1) &(k−1)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=(2k−1)k

((k−1)2+1)((k−1)2+1+k)and

bk=1

k2.Now,

lim

k→∞

ak

bk

=lim

k→∞

(2k−1)k

((k−1)2+1)((k−1)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

(2k−1)k

((k−1)2+1)((k−1)2+1+k)is convergent. Hence "∞

k=1"k2

n=(k−1)2+1

k

n(n+k)is also conver-

gent by the comparison test because we saw before that

∞

!

k=1

k2

!

n=(k−1)2+1

k

n(n+k)≤

∞

!

k=1

(2k−1)k

((k−1)2+1)((k−1)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]Studentdifﬁcultiesunderstanding

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

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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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