One of my favourite mathematical results is the famous formula
As far as I’m concerned, all of maths is here, and if this formula
doesn’t blow you away then you simply have no soul. What the formula
does is to connect two quite different concepts, the geometry linked to
the number
and the simplicity of the odd numbers. The result is truly magical and
surprising, and exactly illustrates the extraordinary way that maths can
link patterns together. Whenever I am asked to define mathematics I
simply write down equation (1). Those of you who think that maths is
just a language, think again.
How quickly do these series converge to a value involving π?
This formula has a splendid history. It was derived in the West in 1671 by James Gregory from the formula for arctan(x) and slightly later and independently by Gottfried Leibniz.
However, the same formula (along with many other results involving
infinite series) was discovered long before in the 1300s by the great
Indian mathematician Madhava. Similar results of equal beauty are the convergent series given byThe world needs pi
So what, you (and many pure mathematicians) might say. Surely you don’t need to know the value of that accurately, after all the Bible was content to give it to just one significant figure. However, is not any number. It lies at the heart of any technology that involves rotation or waves, and that is much of mechanical and electrical engineering. If rotating parts in, say, a typical jet engine are not manufactured to high tolerance, then the parts simply won’t rotate. This typically involves measurements correct to one part in and, as these measurements involve , we require a value of to at least this order of accuracy to prevent errors. In medical imaging using CAT or MRI scanners, the scanning devices move on a ring which has to be manufactured to a tolerance of one part in , requiring an even more precise value of .However, even this level of accuracy pales into insignificance when we look at modern electrical devices. In high frequency electronics, with frequencies in the order of 1GHz (typical for mobile phones or GPS applications), electrical engineers have to work with functions of the form where and is a number close to one. To get the accuracy in the function needed for GPS to work requires a precision in the value used for in the order of one part in .
So, to live in the modern world we really do need to know very accurately.
So, what can we do? One possibility is to take a vast number of terms of the series for etc. above, book lots of time on a very expensive computer, sit back and wait (and wait, and wait). Or we can try and accelerate their convergence. So that with only a small number of terms (say 10) we can get 10 significant figures for . The nice thing about this method is that the derivation of the formulae is very transparent (well within the reach of a first year undergraduate or even a good A-level student). In principle this method can also be used to find the sum of other slowly convergent series.
Accelerating the convergence of a series
Let’s suppose that we have a seriesTo illustrate this idea we will take series (2) for . Let’s define
Figure 1: The sum Sn which monotonically increases to π 2/6.
Suppose that we next look at the difference . A plot of is given in figure 2(a) and we can see that this decreases to zero as increases. But how fast?To estimate this we plot in figure 2(b) the values of . It is clear from these figures that while as , approaches a constant that looks suspiciously like one.
Figure 2: (a) The error En tending to zero. (b) nEn which tends to one.
From this, we might guess that so that to a first approximationBut what are the Br?
It turns out that the Br are a very well-known set of numbers. There is a nice and systematic way to calculate their values (but if you want you can skip this calculation and go straight to the result).If we take (6) and replace by then we get
Now, we can combine these expressions with the equation (9). This gives
Jakob Bernoulli, 1654 – 1705.
These numbers are famous and are called the Bernoulli numbers. They come up everywhere, from number theory to mechanics and beyond. Jakob Bernoulli described them in the book Ars Conjectandi
(published posthumously in 1713) in connection with sums of powers of
integers and they were discovered almost simultaneously by a number of
other mathematicians. Since then, they have played a starring role in
mathematical history. For example they are very important in the
understanding of both Fermat’s last theorem and the Riemann zeta function, and the first computer program written by Ada Lovelace in 1842 was designed to compute them.Doing a quick sum
Putting this all together, for any fixed values of and we can add up terms of the original series to find and then add up terms of the series
to get the correction. We can then estimate by
The error between this approximation and is approximately given by the next term in the series which is given by For example, if then
where the last term expresses the fact that the error we expect to see is proportional to . For terms this would mean an error of which is a huge improvement over the error of about that we would get from naively adding up the series.
So, what’s the catch with doing this? Well the problem with this approach is that as the Bernouili numbers increase rapidly as increases, then ultimately the correction term gets rather large. In fact if is fixed then the individual terms in tend to infinity as . For an optimal error for a given value of it generally works best to truncate the series for the correction and to take , estimating by . This gives an error proportional to We can easily play the same game with the other series mentioned in this article. See this appendix for details.
How well does this work?
Very! To show this we will look at a table of values of for and , and then the three corrections given by1 | 1 | 2 | 1.666666666666667 | 1.623809523809524 |
5 | 1.463611111111111 | 1.663611111111111 | 1.644944444444445 | 1.644934065473016 |
10 | 1.549767731166541 | 1.649767731166541 | 1.644934397833208 | 1.644934066847493 |
Is this new?
Not at all. The basic idea of accelerating the convergence of a series certainly goes back to the extraordinary mathematician Leonhard Euler, if not earlier. Faced with an alternating series, such as the first one (1) that we looked at to calculate in which the terms alternate in sign, Euler devised a transformation, which instead of summing the series, summed the various divided differences of the series. This method does not work for series with all positive terms such as the series (2) to calculate . However, Euler needed to find the sum of this series as at the time (1735) its sum was unknown. The question of finding it (and finding a proof) was posed in 1644, and many distinguished mathematicians had tried, and failed, to find the sum. Euler’s idea was to compare the sum of terms of the form with the integral of the function . As was well known at the timeThe problem was renamed the Basel problem in honour of the home town of both Euler and the Bernoullis, and Euler’s reputation (at the age of only 28) was made for life.
By his method of attack, Euler anticipated a lot of modern mathematics, in which the computer is used as an experimental tool, to gain insight into the solution of a mathematical problem as a first stage in proving the result. Nowadays we use this method all the time, and it continues to rely making highly accurate calculations. But what is nice about the techniques I have described in this article is that we can see that by doing only a little extra mathematics we can do much better than even the most powerful computers.
About the author
He has co-written the popular mathematics book Mathematics Galore!, published by Oxford University Press, with C. Sangwin.
0 komentar:
Posting Komentar