Infinity and Arithmetic ProgressionsAuthor(s): Dave Hewitt and A 2-year Mathematics PGCE GroupSource: Mathematics in School, Vol. 24, No. 3 (May, 1995), pp. 34-35Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30228207 .

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INFINITY and

arithmetic rogressions

by Dave Hewitt and a 2-year mathematics PGCE group

The following work took place with a group of students on a two-year PGCE conversion course, the first year of which students develop their own mathematics. Descriptive com- ments of the session are accompanied by some written work the students did following the session and some reflections of my own. All reflections are printed in italics. The group had already done some work with arithmetic progressions where they had generated formulae to find the sum of a general arithmetic progression.

I put the following on the board:

1+3+5+7+ ...

In this case the series has an infinite number of terms and it was clear that the sum of this would also be infinite. I asked whether they could find an infinite arithmetic progression whose sum was finite. Although it failed to meet my request for an infinite number of terms, Jason offered the following:

-00+0 o +00

+ co

This had an initial term of - oo and a common difference of + co. There was some laughter about this example but no-one challenged the statement that - oo + co =0. So I decided that I would play devil's advocate. I offered two sequences, one of which gives - oo and the other giving + 00o:

-oo0= -1+-2+-3+-4+-5+-6+ ...

oo = 2 +4+6+8 + 10 + 12+ ...

Then I added them together by saying that each term in the second expression cancels one of the terms in the first expression. So, I am left with:

- co+oo= -1 +(-2+2)+ -3+(-4+4)

+ - 5 + (-6+6)+ ..

=-1+-3+-5+ ..

= - oo

I claimed that - oo + co = - co, and not zero as Jason had suggested. Cathy then said that you could add the two series up in a different way, by adding the first term of each together, then the second terms, then the third, and so on. This would give:

- co + o = (-1 +2) +(- 2 + 4) + (- 3 + 6)

+ (-4 + 8) +(-5 + 10) + ..

=1+2+3+4+5+6+ ...

=-00

And so Cathy claimed that - co + 00 = co. So, we now had:

- oo + oo = 0 (Jason)

- 00 + 00 = - 00 (myself)

- oo + oo = co (Cathy)

This was left unresolved!

I did not enjoy the discussion on infinity. Ifeel mathematically secure with definitions, formulae, proofs, lemmas, solutions and the use of q.e.d. I like the exactness of mathematics. Infinity cannot be neatly summarised and the discussion of infinity does not seem to hold the promise of definitive statements. On reflection: am I hung up on an answer- orientated approach to mathematics? (Duncan)

The discussion in class began to blur the edges of my new personal definitions and ideas concerning A.P.s, so I quickly switched off. I listened to the discussion, but meanwhile I concentrated on not losing sight of the clear picture of what I had learned previously. (Richard)

There is a dilemma I occasionally feel between deliberately avoiding questions which might challenge someone's ideas,

34 Mathematics in School, May 1995

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and deliberately considering such questions. I know that I am not interested in looking at mathematics superficially, and that "big" challenging questions can be found in any area of mathematics and at any "level" of mathematics. As a consequence my students can end up having more questions at the end of a session than they had at the beginning. They can arrive feeling that they understand something and leave feeling as if they do not. I know that the more I learn about mathematics, I realise that what I consider to be simple is in fact immensely complex. Mathematics is complex, should I pretend it is otherwise to my students? What is engaging about mathematics is its complexities, the rest is just routine. (DH)

Someone made a statement suggesting that an infinite arithmetic progression always tended to positive or negative infinity. I asked whether that statement could be carefully worded. The first attempt was:

Conjecture 1: An arithmetic series will always tend to positive or negative infinity and will not converge to a finite value.

It took four amendments before the group agreed a final version:

Conjecture 1: The sum of an infinite arithmetic series will always tend to positive or negative infinity and will not converge to a finite value, except when the first term is zero and there is a common difference of zero.

There followed a second conjecture, which had two amendments before agreement was reached:

Conjecture 2: The terms of an infinite arithmetic series will always tend to positive or negative infinity, except when there is a common difference of zero.

Thinking of the two conjectures, i.e. important characteristic propreties of arithmetic series believed generally to be true, did not prove to be too problematical: having formed the conjectures, modifications were made to include exceptions which came to mind. The only frustrating thing about forming the conjectures was the fact that perhaps there were exceptions which didn't come to mind, or perhaps some key point has been left out. Here lies the advantage of working in groups: brainstorming ideas is an invaluable technique to covering many alternatives. (Steve M)

Groups were asked to work on a proof of either of the two conjectures. Several of the proofs they produced involved placing

xo into a formula they had for the sum of a finite

arithmetic progression. For example, Shaf used the formula for the nth term in

an arithmetic sequence:

nth term = a + (n - 1)d

where a is the first term and d is the common difference. He placed oo into the formula and wrote the following

with respect to Conjecture 2:

a + (oo - 1)d tends to oo since oo- 1 is still co,

and oo x d tends to co.

Therefore, a + (oo - 1)d= a + oo which tends to oo. Becky was aware that in her working there were a

number of assumptions made. For example:

20 -=o oc oo oo x d= oo

2

What are the problems we encounter when trying to write proofs ourselves? My experience ... is the temptation to include in our proof as justification, just the mathematical

Mathematics in School, May 1995

fact we are trying to prove! Another problem is knowing where to start, i.e. a point from which, when expressing ourselves mathematically, we will lead systematically to our objective. (Ashley)

Some people considered the case when d= 0, and assumed that anything times zero is zero. On the board I wrote:

0 x 0o

i.e., (something which tends to 0)

X

(something which tends to co)

I was offered 1/n for the first, and n for the second. We considered what happens as n tends to infinity:

1 -x2=l 2

1

-x3=1 3

1 - x4=1 4

xn= n

Every term of this sequence is one whilst n tends to infinity. Several people realised that other examples could be generated to produce different answers to the question of what 0 x oo was.

Jason linked the earlier issue of - co + co with the issue of 0 x 0o:

00o + - co =(1 x o) +(-1 x oo)=(1 + -1) x 00 = x00

Paul wrote the following:

If we interpret 0 as "a very small number" and cc as "a very large number", then: 0 x o = "a very small number" x "a very large number" This could indicate any of a range of finite numbers.

Gordie wrote:

We can say that d x co = 0 if d= 0, because d will be a finite value of zero and will not be tending to zero. It would make a huge difference if d was tending to zero in the form of 2, '~, 1,-i, ... in this case we would have:

1 1 xc0=1.

oo

Disadvantages of rigour: Rigorous treatment can be difficult to understand; Likely to be confusingfor newcomers to a subject.

Advantages of rigour: Being rigorous forces you to understand what may be ambiguous in a situation; Understanding of a subject is reinforced; It can help to look at a subject from a fresh viewpoint. (Colin)

Infinity seems to pose questions by its mere existence, and students can find their own ways of dealing with big questions so that they can use the notion of infinity.

Caleb Gattegno talked of mathematics being shot through with infinity, and the last quote is from Andrew: Often all the magic seems to occur at infinity. If I combine these two quotes, can I help my students find some "magic" within any area of mathematics?*

35

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