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 by
and
However, in one sense, these formulae are disappointing. If you want to actually calculate

or

then you probably would not reach for one of these formulae. The reason
is that they converge very slowly. If you take formula (1), add up 100
terms and multiply by 4, you get 3.146567747182956, which whilst fairly
close to

is not a particularly accurate estimate given the effort involved in adding up 100 terms. If you wanted to calculate

or

to an accuracy of six decimal places, you would have to take on the order of

terms of either series (1) or (2), and long before you have added up
all of the terms in the series, the rounding errors associated with
computer calculations will have accumulated to the point where the
accuracy of the answer is severely degraded.
The 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 series
and we define the sum

by
We will assume that this series
converges. This means there is a
limiting sum 
so that
If we want to work out the value of

then we can simply take the values

and let

get very large. However, just adding up a series loses quite a lot of
the information contained within it, and is really a very crude thing to
do. Maybe we can squeeze more information out of the series and use
this information to accelerate the convergence of

. This means that we add a correction term to

so that it approaches

much more rapidly. What is nice about this approach is that it is quite easy to work out the correction terms.
To illustrate this idea we will take series (2) for

. Let’s define
As

increases,

also increases steadily towards the value of

. In figure 1 we plot the values of

which you can see are increasing towards the value of

.

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 approximation
We can improve on this guess by assuming that there is a sequence of numbers

such that as
If we can calculate the terms

for

then we can estimate

from the value of

by the expression
But 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
so that
Also, from the definition of

we know that
Combining these two expressions we get
We can now find the values of each of the terms

recursively by expanding each of these expressions in powers of

and considering

to be very large. For example
and
More generally
where
These are the so-called
Taylor series expansions for the above expressions.
Now, we can combine these expressions with the equation (9). This gives
To find the value of each of the terms

we compare the expressions involving terms of the form

for

. For example, the coefficient of

on the left of the above equation is 1 and on the right it is

. Hence
If we next look at the terms in

and

we get (respectively)
Substituting the value of

into the first equation gives

and knowing this we can then find

from the second equation. This gives
We can then continue inductively, and can work out the terms

from the recurrence relations
Turning the handle we get the following numbers
We notice that if

is odd and bigger than one then

, and that the values of

with

even alternate in sign. The numbers that we have calculated are all quite small, but they get much bigger (rapidly!) as

increases. For example

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 by
 |
 |
 |
 |
 |
1 |
1 |
2 |
1.666666666666667 |
1.623809523809524 |
5 |
1.463611111111111 |
1.663611111111111 |
1.644944444444445 |
1.644934065473016 |
10 |
1.549767731166541 |
1.649767731166541 |
1.644934397833208 |
1.644934066847493 |
We can see that the approximation

is a very good approximation indeed for

with an error of only

. This is especially impressive when we see that

is a very poor approximation to

. Note that we have only had to add in an extra

terms to do this. To gain the same level of accuracy with

we would have to have included another

terms! Readers are invited to work out

and

to further test the impressive accuracy of this method.
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 time
and the sum of the terms

is an approximation to this integral. If you can find the error in this
approximation then you can sum the series by comparing it to the known
integral. In a (typical) tour-de-force, Euler was able to calculate this
error deriving the (now famous) Euler-Maclaurin (divergent) series for
the error. If you take

terms of the original series for the sum and

terms of the Euler-Maclaurin series then you get exactly our formula
(13). It is pleasing (for a numerical analyst like me) to recount that
Euler used his formula to calculate the sum of the series (2) correct to
20 decimal places (without of course the benefit of any form of
caculator). From this calculation he was then able to
guess that the sum of the series was

. Knowing the answer he was then able to find a proof.
The 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
Chris Budd is Professor of Applied Mathematics at the University of Bath, Vice President of the
Institute of Mathematics and its Applications, Chair of Mathematics for the
Royal Institution and an honorary fellow of the
British Science Association. He is particularly interested in applying mathematics to the real world and promoting the public understanding of mathematics.
He has co-written the popular mathematics book
Mathematics Galore!, published by Oxford University Press, with C. Sangwin.