by Wells, David
The remarkable feature of this definition is that it captures everything that we want to say about the infinite sum being 1, without actually talking about infinity at all! The challenge you set me involves a finite number, albeit very small. I answer by showing that a finite number of terms will take the partial sum to within less than the number you named. Infinity has vanished, which is one reason why we can be confident that our definition is solid and game-like, rather than vague, poetic and fishy.
Figure 8.3 Parabola
That trick doesn't solve all the problems of infinity by any means. Here is a geometrical puzzle about infinity: of the three conic sections, the ellipse is finite but both the parabola and hyperbola both ‘go off to infinity’ although in seemingly different ways. The parabola (Figure 8.3) looks like an ellipse which has been stretched for ever while the hyperbola (Figure 8.4) gets closer and closer to its two asymptotes as it disappears to infinity.
Figure 8.4 Hyperbola with asymptotes
How can we formalise these very vague and intuitive ideas? Figure 8.5 is a picture of a parabola as a closed curve touching a line. What does this mean? What should it mean?
Figure 8.5 ‘Parabola’ touching line at infinity
In projective geometry, a line at infinity is added to the Euclidean plane, containing an infinity of points at infinity. The parabola is then a conic which touches the line at infinity at a single real point which is also on its axis, while the hyperbola, together with its asymptotes, intersects the line at infinity in two distinct real points. (And all circles intersect the line at infinity in the same two imaginary points, but that's another story.)
Calculus and the idea of a tangent
Tangent is another easy-to-understand idea that is tricky to pin down. Small children can use a ruler to draw a line which touches a circle but to make it touch ‘exactly’ at a certain point is not so easy. It becomes easier when you realise that the tangent of a circle – by symmetry! – ought to be perpendicular to the radius, but that leaves the problem of drawing accurate tangents to curves like the parabola in Figure 8.6.
Figure 8.6 Parabola with tangent
This is the parabola y = x2, drawn with horizontal and vertical scales in the ratio 2:1. What is the slope of a tangent touching the curve at (1,1)? Draw the tangent roughly and it seems to be about 2, but drawing is no proof. The set of chords in Figure 8.7 is also suggestive: they are parallel, they have slope 2 and they appear to be parallel to the tangent at (1,1).
Figure 8.7 Parabola with tangent and parallel chords
For another natural hint let's calculate the slopes of the following sequence of regularly spaced chords through (3,9) (Figure 8.8).
Figure 8.8 Parabola with chords through one point
The sequence of chords from AG to GG have slopes, AG
BG
CG
DG
EG
FG
‘GG'
0
1
2
3
4
5
?
which strongly suggests that the tangent – which ‘joins the point (3,9) to itself’ – should have slope exactly 6, but how can we ‘prove’ this? Once again, we need a definition that matches the situation.
The slope of the chord joining (3,9) to (t,t2) is,
by basic algebra, so when t = 3, the slope will indeed be 6. In other words, we imagine the chord rotating as (t,t2) moves closer and closer to (3,9) until in the limit the chord doesn't actually exist because you can't draw a line joining a point to itself, but the slope of the chord in the limit is 6. Then we take the final deliberate step and define the tangent to be the line through (3,9) with slope 6.
Notice all the steps in this process: we start with children making actual physical experiments complete with inevitable experimental errors, on the understanding that a tangent is a straight line that ‘touches’ a curve, whatever that means; we move on to a parabola and our own calculations of the slopes of chords, still without any clear idea of what a ‘tangent’ is or ‘ought to be’; finally, we get a simple answer that ‘makes sense’ and answers all our doubts – and we then turn round and define a tangent to be the answer we got!
This is not a circular argument because we haven't actually gone round in a circle. Rather, we started like a detective with some strong clues as to the kind of object a tangent ought to be. We experimented to find more clues, and finally we had enough clues to point us with confidence towards the solution.
Actually, this is not the only possible definition. Let's go back to the parallel chords which are indeed parallel to the tangent. These chords all meet the parabola in two distinct points. If we recall that algebraic equations – including quadratics – can have double as well as single roots, then we might hope that the tangent at (3,9) will prove to have a double intersection with the parabola at that point. The equation of the line through (3 − t, (3 − t)2) and (3 + t, (3 + t)2) is,
This will intersect y = x2 when x2 − 6x + (9 − t2) = 0 and this does indeed have a double root when t = 0, as expected. Once again, we are home and dry because all these chords have slope 6 so they are parallel, and they turn out to be parallel to the tangent at (3,9), according to our second new definition which defines the tangent algebraically without talking about limits at all: the tangent at (3,9) is the line which when we solve the equation to find where it meets the parabola, gives a double root.
(This doesn't explain why there could not be more than one tangent at one point: occasionally there is. We also need to prove that these two definitions of a tangent usually give the same answer and to understand any exceptional cases when they do not.)
Mathematical textbooks are much more precise in defining a tangent than we have been: a general definition must apply to a general curve, or at least a wide class of curves that ideally includes all the curves we are interested in. Are there ‘curves’ for which these definitions don't work? Yes, there are weird ‘curves’ which seem hardly ‘curves’ at all because they do not have tangents. Naturally, mathematicians have investigated them also – and so maths moves on, ever forwards and upwards!
Will the informal and psychologically powerful concept of infinity ever be completely tamed? Plausibly not, because ideas of infinity – ‘ideas’ in the plural – turn up so often in so many different parts of the mathematical landscape.
What is the shape of a parabola?
We conclude by looking at another example of a familiar term being less clear than we suppose, also illustrated by the parabola.
The idea of ‘shape’ might seem pretty unproblematic. All circles are obviously the same shape, all squares are the same shape, all triangles are not, and parabolas and ellipses vary from narrow to broad or from nearly-circular to long and narrow (Figures 8.9 and 8.10).
Figure 8.9 Two parabolas, different ‘shapes’
Figure 8.10 Two ellipses, different ‘shapes’
It is also easy to convince yourself by experiment, or reasoning with a diagram on a square grid, that if you ‘blow up’ a figure from a centre of similarity, O, the new figure has the same shape. The pentagon in Figure 8.11 has been ‘blown up’ by a factor of 2, meaning that each new point, A’, B’, C’…is twice times as far from O as its original point.
Figure 8.11 Two pentagons in perspective
The claim, however, that parabolas vary from narrow to broad is quite misleading and illustrates how even apparently simple ideas like shape must be treated with kid gloves. There is a case for arguing – against first appearances – that all parabolas are the same shape.
Figure 8.12 Two parabolas with ‘matching’ chords
In Figure 8.12 one parabola has been blown up by a factor of 2, from its vertex, to create 6 new points, plus the original vertex, lying on a new parabola. According to our previous argument the resulting figure ought to be same shape. Indeed, the resulting 7 points do lie on another parabola which seems, however, to be ‘wider’ than the orig
inal parabola. So is it the same shape or not?
Searching for a decisive perspective we can resort to the equation of the parabola, y = ax2. Changing y and x by the same factor c, so that the shape is unchanged, we get the equation
Since c can be any number, it does seem that any two parabolas, y = ax2 and y = acx2 have the same shape. Still not convinced? Then let's draw the same portion of the ‘narrow’ and ‘broad’ parabolas next to each other. Now they do look genuinely to be the same shape but of different sizes which is a clue to what is happening. The whole of any parabola can be mapped on to the whole of any other parabola, by expanding one or contracting the other. So whole-parabolas are all the same shape but parts-of-parabolas are not the same shape (Figure 8.13).
Figure 8.13 Two parabolas with matching points
Conversely, we can easily find parts of two rectangles, however different their whole-shapes, that are the same shape. The square and rectangle in Figure 8.14 are obviously not the same shape, but the parts cut off below are similar.
Figure 8.14 Two rectangles
If even shape is not such a simple idea what other everyday ideas might prove to be mathematically tricky?
9 Convergent and divergent series
The pioneers
Archimedes (287–212 BCE) was the first (known) genius in the history of mathematics. Among his many achievements he anticipated modern calculus and analysis by finding the areas of curves and the volumes of solids with curved surfaces. He was the first mathematician to sum an infinite series in his book On the Quadrature of the Parabola.
To find the area of the parabolic segment cut off by the line AB he drew the tangent, CC′ which is parallel to AB, to create the triangle ABC (Figure 9.1).
Figure 9.1 Parabola with tangent and triangle
He then drew the tangents at D, parallel to AC and at E, parallel to BC, to create the triangles ACD and BCE (Figure 9.2), and proved that the area of each of the two small triangles is 1/8 of the main triangle.
Figure 9.2 Parabola with three triangles
This is already very clever, but now Archimedes produced his master stroke: he imagined drawing the tangents parallel to AD, DC, CE and EB to create 4 more triangles, and then 8 more triangles, and so on. In this way, he argued that the area of the parabolic segment cut off by AB was the sum of all these triangles because when you added all their areas together you would have finally exhausted the total area of the parabolic segment.
If we call the area of the triangle ABC, 1, then the total area of the parabolic segment is the sum for this infinite series:
Archimedes got the answer we would expect, 4/3, which was a mystery to other Greek mathematicians who had no idea how an infinite series could have a finite sum.
The next remarkable figure in the history of infinite series was the Indian mathematician Madhava who was born in 1350 at Sangamagramma in Kerala and died in 1425, almost exactly three hundred years before Sir Isaac Newton (1642–1727) and other mathematicians who pioneered infinite series in Europe.
Madhava developed infinite series to represent functions, and he may well have discovered series equivalent to our series for sin x, cos x and arctan x around 1400. Madhava's original works have all been lost, but their substance was preserved by his followers in the flourishing school he founded. Thus when Jyesthadeva wrote his Yukti-Bhasa around 1550 he explained one of Madhava's series which is equivalent to James Gregory's series,
When Madhava put x = π/4 he got the series which we call Leibniz's series, but which would more justly be named after Madhava:
Putting x = π/6 he got the series,
This may be the series that he used to calculate π to 11 decimal places: π = 3.14159265359…[Madhava: MacTutor]. European mathematicians did eventually catch up. Leibniz used the newly discovered calculus that he had invented at the same time as Newton to find the series for sin x, and cos x:
and
and James Gregory (1638–1675) rediscovered the above series for arctan x. Newton then discovered the binomial theorem for fractional indices, by analogy with the finite expansion of (x + y)n when n is an integer, and the Europeans were off – but long after Madhava and the Kerala school had first studied infinite series. (The history of mathematics is not as simple as once thought, and not as purely European.)
The harmonic series diverges
Nicholas Oresme (1323–1382) has been claimed as the inventor of graphs and coordinates which he used to plot one dependent variable against an independent variable, just as schoolchildren learn to do today [MacTutor: Oresme]. He had no general interest in infinite series but he did prove that the harmonic series,
does not have a sum. It might seem obvious that it must converge to a limit because the terms are getting steadily smaller. Indeed they are, but not fast enough. To see why it diverges Oresme compared it with another series which definitely diverges:
Every term of the harmonic series is equal to or greater than the corresponding term in the second series, all of whose terms can be gathered together like this:
The bracketed groups of terms all sum to 1/2 and the brackets go on for ever so we can add 1/2 as many times as we wish to the running total by adding an extra 2, 4, 8, 16, 32…terms. Because we have to add more and more terms to increase the sum by just 2, it does diverge very slowly indeed, but diverge it does and so does the harmonic series. How slowly? To get beyond 4 we have to add 31 terms, to get beyond 10, we need 12367 terms, and more than 250000000 terms to pass 20 [Sloane A004080].
There is another very different way to show that the harmonic series diverges. Suppose that it converges, and that its sum is S. (This turns out to be another case of talking about something which doesn't actually exist.)
Then,
But 1/2+1/4+1/6+1/8+⋯ = S, and 1/1+1/3+1/5+1/7+⋯ is greater than S because each corresponding term is greater. So it follows that S = S + a sum greater than S, which is impossible. So the series cannot be convergent – it must be divergent.
Weird objects and mysterious situations
Divergent series – the harmonic series was the first and simplest of many – may be amusing curiosities but if they have no sum, what use can they be? One response to the paradox was to abandon them as best avoided, which is exactly what some early modern mathematicians did. Fearful of getting wrong answers, they shied away.
Euler, however, was not deterred. He set his course boldly straight ahead and produced typically astonishing results. Indeed, astonishing is hardly the word, absurd might seem a better adjective. Yet Euler kept going because the results he got were consistent, they made some kind of sense, and so he understood his problem as working out the meaning of divergent series. Let's see what happened in practice, starting with a very simple series that oscillates. The series,
can be seen as, (1 − 1) + (1 − 1) + (1 − 1) + (1 − 1) + ⋯
which is apparently zero, but also as,
which seems to be 1. Leibniz argued that since the partial sums when you start adding up the terms from the start, are 1, 0, 1, 0, 1, 0…with equal probability, then the sum ‘should be’ 1/2. This certainly shows imagination in bringing chance and probability into the sum of a fixed series [Kline 1983: 308].
Guido Grandi, a mathematician who was also a Jesuit priest, drew a different and mystical conclusion from this ‘demonstration’ that 0 = 1: it showed mathematically, he claimed, how the world could have been created from nothing [Knopp 1928: 133n].
Euler was smarter, and more mathematically imaginative: he took the series,
and compared it to the series we have already met:
Putting x = −1, he concluded that the sum of the first series ought to be 1/2, which agrees with Leibniz but still seems absurd. James Bernoulli in 1696 described it as a ‘not inelegant paradox’. It was indeed, but paradoxes exist to be explained and Euler's basic idea that you should interpret divergent series as specific values of a function was profound and extremely powerful. Euler had confidence in his own judgement and intuition
, and went on to conclude that,
In a later paper, On Divergent Series (1760) he discussed what it means to find the ‘sum’ of a series and promised ‘to clarify a concept causing up to now the greatest difficulties’ [Barbeau 1979: 357]. He then looked at the series,
With typical imagination and consistency rather than rigour, he used several different methods to get different results for its sum, including 0.580, 0.60542, and 0.59966 [Barbeau 1979: 357] [Euler 1760a].
