Convergence acceleration of series
(Click here
for a Postscript version of this page.)
1 Introduction
Numerous mathematical constants are calculated as limit of series and many
of those series are very slow to converge requiring therefore methods to
accelerate their convergence. In this section, we recall some definitions
and elementary results on well known series.
For any series åkak, we denote by sn its partial sums, that
is
|
ê ê ê
ê ê ê
|
|
|
sn=a0+a1+...+an= |
n å
k=0
|
ak. |
|
|
|
|
We are interested in the behavior of sn as n tends to infinite (that
is infinite series). The series whose terms have alternative signs
are said to be alternating series, we write them as
sn=a0-a1+...+(-1)nan= |
n å
k=0
|
(-1)kak, |
|
where the (ak) are positive numbers.
Definition 1
An infinite series å ak is said to be convergent if its partial sums
(sn) tends to a limit S (called the sum of the series) as n
tends to infinite. Otherwise the series is said to be divergent.
The following examples are elementary and occur frequently and it's usual to
compare a given series to one of those.
Example 2
(Geometric series) For the geometric series which is defined by ak=xk, the partial sum sn is then given by
sn=1+x+x2+...+xn= |
1-xn+1
1-x
|
, |
|
hence it converges for | x| < 1 to S=1/(1-x) and diverges
otherwise.
Example 3
(Harmonic series) The harmonic series is defined for k > 0 by ak=1/k, and the partial sum sn (sometimes denoted Hn and
called Harmonic number) is
sn=1+ |
1
2
|
+ |
1
3
|
...+ |
1
n
|
. |
|
We have for n=2p
|
|
|
æ è
|
1+ |
1
2
|
ö ø
|
+ |
æ è
|
1
3
|
+ |
1
4
|
ö ø
|
+ |
æ è
|
1
5
|
+..+ |
1
8
|
ö ø
|
+...+ |
æ è
|
1
2p-1+1
|
+...+ |
1
2p
|
ö ø
|
|
| |
|
1. |
1
2
|
+2. |
1
4
|
+4. |
1
8
|
+...+2p-1. |
1
2p
|
= |
p
2
|
, |
|
|
therefore the partial sums are unbounded and the harmonic series is
divergent (this important result was known to Jakob Bernoulli (1654-1705) and much before to the french bishop Nicolas Oresme (1323-1382)).
Example 4
(Riemann Zeta series) A natural generalization of the harmonic series is to
consider the partial sum
sn=1+ |
1
2s
|
+ |
1
3s
|
+...+ |
1
ns
|
|
|
which converges to the Riemann Zeta function z (s) for any complex
numbers s such as  (s) > 1. The case s=1 is the Harmonic series and
the limits for s=2p are well known since Euler who established for example
that z(2)=p2/6, z(4)=p4/90,...
2 Kummer's acceleration method
Ernst Kummer's (1810-1893) method is very natural to understand and may be
used to accelerate the convergence of many series. It goes back to 1837 and
the idea is to subtract from a given convergent series åak another
equivalent series åbk whose sum C=åk ³ 0bk is
well known. More precisely suppose that
|
lim
k®¥
|
|
æ è
|
ak
bk
|
ö ø
|
=r ¹ 0, |
|
then the transformation is given by
|
¥ å
k=0
|
ak=r |
¥ å
k=0
|
bk+ |
¥ å
k=0
|
|
æ è
|
1-r |
bk
ak
|
ö ø
|
ak=rC+ |
¥ å
k=0
|
|
æ è
|
1-r |
bk
ak
|
ö ø
|
ak. |
| (1) |
The convergence of the latest series is faster because 1-rbk/ak
tends to 0 as k tends to infinity [1].
Example 5
Consider
|
|
| |
|
|
¥ å
k=1
|
|
1
k(k+1)
|
= |
¥ å
k=1
|
|
æ è
|
1
k
|
- |
1
k+1
|
ö ø
|
=1, |
|
|
we have r = C=1 and Kummer's transformation (1) becomes
S=1+ |
¥ å
k=1
|
|
æ è
|
1- |
k2
k(k+1)
|
ö ø
|
|
1
k2
|
=1+ |
¥ å
k=1
|
|
1
k2(k+1)
|
. |
|
The process can be repeated with this time
|
|
| |
|
|
¥ å
k=1
|
|
1
k(k+1)(k+2)
|
= |
1
4
|
, |
|
|
giving
S= |
5
4
|
+2 |
¥ å
k=1
|
|
1
k2(k+1)(k+2)
|
. |
|
After N applications of the transformation
S= |
N å
k=1
|
|
1
k2
|
+N! |
¥ å
k=1
|
|
1
k2(k+1)(k+2)...(k+N)
|
, |
|
whose convergence may be rapid enough for suitable values of N (this
example is due to Knopp [8]). Note, that we have used the following
family of series to improve the convergence
|
|
|
¥ å
k=1
|
bk= |
¥ å
k=1
|
|
1
k(k+1)...(k+N)
|
= |
1
N.N!
|
, |
| |
|
|
|
3 Richardson Extrapolation method
Suppose that the partial sum verifies the relation
S=sn+ |
a
np
|
+O |
æ è
|
1
np+1
|
ö ø
|
, |
| (2) |
where S is the limit of the sum, p a known integer and a a
constant which we don't need to determine. Then, following the ideas of
Lewis Fry Richardson (1881-1953) [10], it's natural to define
the transformation
sn(1)= |
2ps2n-sn
2p-1
|
=s2n+ |
1
2p-1
|
(s2n-sn) |
|
in order to eliminate the term in a from relation (2). The process may be repeated on the new series
|
|
|
2p+1s2n(1)-sn(1)
2p+1-1
|
, |
| |
|
|
2p+2s2n(2)-sn(2)
2p+2-1
|
, |
| |
|
|
|
Example 6
Suppose that you have computed, by geometric means and like Archimedes did,
the perimeters of the circumscribed polygon with respectively 6,12,24,48
and 96 sides on a unit circle then in this case S=p, p=2 and
Richardson's process becomes here
|
|
| |
|
s2n(1)+ |
1
15
|
(s2n(1)-sn(1)) , |
| |
|
s2n(2)+ |
1
63
|
(s2n(2)-sn(2)) , |
| |
|
|
|
It produces the approximations (the number of sides is 6n)
and finally we find p » s1(4) =3.141592653(637...) which
is correct to nearly 10 digits and therefore much more accurate than the 2
digits value estimated by Archimedes !
4 Aitken's acceleration and related methods
4.1 Aitken's d2-process
In 1926, Alexander Aitken (1895-1967) found a new procedure to accelerate
the rate of convergence of a series, the Aitken d2-process [2]. It consists in constructing a new series S(1),
whose partial sums sn(1) are given by
sn(1)=sn+1- |
(sn+1-sn)2
sn+1-2sn+sn-1
|
= |
sn-1sn+1-sn2
sn+1-2sn+sn-1
|
, |
| (3) |
where (sn-1,sn,sn+1) are three successive partial sums of the
original series. Of course, it's possible to repeat the process on the new
series S(1) to obtain S(2)... This process was in fact known to
the Japanese mathematician Takakazu Seki Kowa (1642-1708) who used it to
compute 10 correct digits of the constant p by accelerating the
convergence of the polygon algorithm.
For example for the converging geometric series with 0 < x < 1:
|
|
1+x+x2+...+xn= |
1-xn+1
1-x
|
and the limit is |
| |
|
|
|
hence
and if we replace (sn-1,sn,sn+1) by this expression in (3) we have sn(1)=S ..., the exact limit is given just
by three evaluations of the original series. This indicates that for a
series almost geometric, Aitken's process may be very efficient.
Example 7
If we consider the extremely slow (logarithmic rate) converging series
S= |
lim
n® ¥
|
sn= |
lim
n® ¥
|
|
æ è
|
n å
k=0
|
|
(-1)k
k+1
|
ö ø
|
=log(2), |
|
The following array illustrates the result of three consecutive applications
of Aitken's process to the 7 first computed terms of this series :
A formal expression may be obtained in this case ; the expressions of (sn-1,sn,sn+1) are
sn=sn-1+ |
(-1)n
n+1
|
,sn+1=sn+ |
(-1)n+1
n+2
|
|
|
so that
sn(1)=sn+1+ |
(-1)n(n+1)
(n+2)(2n+3)
|
|
|
For n=1000, it becomes
4.2 The e-algorithm
In 1956, Peter Wynn obtained a generalization of Aitken's algorithm which is
called the e-algorithm [11]. It is defined by the
transformation
en(k+1)=en+1(k-1)+ |
1
en+1(k)-en(k)
|
|
| (4) |
starting with en(-1)=0 and en(0)=sn
where sn is the series to be accelerated.
Example 8
Again we take the series
|
lim
n®¥
|
|
æ è
|
n å
k=0
|
|
(-1)k
k+1
|
ö ø
|
=log(2), |
|
|
the following array illustrates the result of consecutive applications of
Wynn's process (4) to the same as in the previous example 7
first computed terms of the series :
and finally en(6)=0.6931524547... Note that the odd values
of the index are intermediate quantities.
5 Euler-Maclaurin summation formula
Suppose that the (ak) may be written as the image of a given
differentiable function f, that is
and so
sn= |
n å
k=0
|
ak= |
n å
k=0
|
f(k) |
|
|
the following famous result due to Leonhard Euler (1732) and Colin Maclaurin
(1742, [9]) states that
sn= |
ó õ
|
n
0
|
f(t)dt+ |
1
2
|
( f(0)+f(n)) + |
m å
k=1
|
|
B2k
(2k)!
|
( f2k-1(n)-f2k-1(0)) +em,n |
|
|
where em,n is a remainder given by
em,n= |
1
(2m+1)!
|
|
ó õ
|
n
0
|
B2m+1(x-
ëx
û )f2m+1(x)dx |
|
|
where the B2k are Bernoulli's Numbers and Bi(x) being Bernoulli's
polynomials (see Bernoulli's number essay for more details or [7]
for a proof).
Example 9
Suppose that s > 1 and
then
|
|
|
| |
|
|
1 s-1
|
|
æ ç
è
|
1- |
1 ns-1
|
ö ÷
ø
|
+ |
1 2
|
|
æ ç
è
|
1+ |
1 ns
|
ö ÷
ø
|
+ |
B2 2
|
|
æ ç
è
|
s
1
|
ö ÷
ø
|
|
æ ç
è
|
1- |
1 ns+1
|
ö ÷
ø
|
+... |
| |
|
+ |
B2m 2m
|
|
æ ç
è
|
s+2m-2
2m-1
|
ö ÷
ø
|
|
æ ç
è
|
1- |
1 ns+2m-1
|
ö ÷
ø
|
+em,n |
|
| |
|
if n® ¥ in this identity, we find the relation
z(s) = |
¥ å
k = 1
|
|
1 ks
|
= |
1 s-1
|
+ |
1 2
|
+ |
B2 2
|
|
æ ç
è
|
s
1
|
ö ÷
ø
|
+...+ |
B2m 2m
|
|
æ ç
è
|
s+2m-2
2m-1
|
ö ÷
ø
|
+em,¥ |
|
and by computing z (s)-sn-1 we find the useful algorithm to
estimate z (s)
z(s) = 1+ |
1 2s
|
+...+ |
1 ns
|
+ |
1 s-1
|
|
1 ns-1
|
- |
1 2ns
|
+ |
m å
i = 1
|
|
B2i 2i
|
|
æ ç
è
|
s+2i-2
2i-1
|
ö ÷
ø
|
|
1 ns+2i-1
|
+( em,¥-em,n) , |
|
and for any given integer m, the remainder tends to zero (Exercise : check
this with care!).
6 Convergence improvement of alternating series
Very efficient methods exist to accelerate the convergence of an alternating
series
one of the first is due to Euler. Usually the transformed series converges
geometrically with a rate of convergence depending on the method.
Because the speed of converge may be dramatically improved, there is a trick
due to Van Wijngaarden which permits to transform any series åbk
with positive terms bk into an alternating series. It may be written
|
¥ å
k=0
|
bk= |
¥ å
k=0
|
(-1)kak |
|
|
with
ak= |
¥ å
j=0
|
2j b2j(k+1)-1=bk+2b2k+1+4b4k+3+8b8k+7+... |
|
|
and because 2j(k+1)-1 increases very rapidly with j, only a few terms
of this sum are required.
Sometimes a closed form exists for the ak, for example if
then
|
|
|
¥ å
j=0
|
2jb2j(k+1)-1= |
¥ å
j=0
|
|
2j
( 2j(k+1)) 2
|
= |
1
(k+1)2
|
|
¥ å
j=0
|
|
1
2j
|
|
| |
|
|
|
and
|
¥ å
k=0
|
|
1
(k+1)2
|
=2 |
¥ å
k=0
|
|
(-1)k
(k+1)2
|
. |
|
6.1 Euler's method
In 1755, Euler gave a first version of what is now called Euler's
transformation. To establish it, observe that the sum of the alternating
series may be written as
2S=a0+(a0-a1)-(a1-a2)+(a2-a3)-...=a0+S1, |
|
|
with
S1=D1a0-D1a1+D1a2-...+(-1)kD1ak+..., |
|
|
and the first difference operator is denoted by
The new series S1 is also alternating and the process may be
applied again
2S1=D1a0+(D1a0-D1a1)-(D1a1-D1a2)+(D1a2-D1a3)-...=D1a0+T1, |
|
|
with this time
T1=D2a0-D2a1+D2a2-...+(-1)kD2ak+..., |
|
|
and the second difference operator is denoted by
the following theorem is then easily deduced by repeating the process (we
omit some details given in [8]).
Theorem 10
(Euler) Let S= åk=0n(-1)kak be a convergent alternating
series, then
S= |
lim
n®¥
|
sn= |
lim
n®¥
|
|
æ è
|
1
2
|
|
n å
k=0
|
|
(-1)k
2k
|
Dka0 |
ö ø
|
, |
| (5) |
and D is the forward difference operator defined by
Dka0 = Dk-1a0-Dk-1a1 = (-1)k |
k å
j = 0
|
(-1)j |
æ ç
è
|
k
j
|
ö ÷
ø
|
aj. |
|
The first values of Dka0 are
|
|
| |
|
| |
|
| |
|
| |
|
a0-ka1+ |
k(k-1)
2!
|
a2- |
k(k-1)(k-2)
3!
|
a3+...+(-1)kak. |
|
|
Example 11
Sometimes it's possible to find a closed form for the transformation (5), take
|
|
|
p
4
|
=1- |
1
3
|
+ |
1
5
|
- |
1
7
|
+...= |
¥ å
k=0
|
(-1)kak with |
| |
|
|
|
the rate of convergence of this famous series is extremely slow, but Euler's
method gives
|
|
| |
|
| |
|
D1a0-D1a1= |
1.2
1.3
|
- |
1.2
3.5
|
= |
1.2.4
1.3.5
|
, |
| |
|
D2a0-D2a1= |
1.2.4
1.3.5
|
- |
1.2.4
3.5.7
|
= |
1.2.4.6
1.3.5.7
|
, |
| |
|
|
|
this suggest the expression (prove it by induction ...)
Dna0= |
1.2.4...(2n)
1.3.5...(2n+1)
|
. |
|
and
|
|
1- |
1
3
|
+ |
1
5
|
- |
1
7
|
+...= |
1
2
|
|
æ è
|
1+ |
1.2
1.3
|
|
1
2
|
+ |
1.2.4
1.3.5
|
|
1
22
|
+... |
ö ø
|
|
| |
|
|
1
2
|
|
æ è
|
1+ |
1
3
|
+ |
1.2
3.5
|
+ |
1.2.3
3.5.7
|
... |
ö ø
|
= |
1
2
|
|
¥ å
k=0
|
|
2kk!2
(2k+1)!
|
, |
|
|
relation also given by Euler.
6.2 Improvement
In 1991, a generalization of Euler's method was given in [4] (this
work in fact generalizes a technique that was used to obtain irrationality
measures of some classical constants like log(2) for example). This
algorithm is very general, powerful and easy to understand. For an
alternating series å(-1)k ak, we assume that there exist a
positive function w(x) such that
This relation permits to write the sum of the series as
|
¥ å
k=0
|
(-1)kak = |
ó õ
|
1
0
|
|
æ è
|
¥ å
k=0
|
(-1)kxk w(x)dx |
ö ø
|
= |
ó õ
|
1
0
|
|
w(x)
1+x
|
dx. |
|
|
Now, for any sequence of polynomials Pn(x) of degree n with Pn(-1) ¹ 0, we denote by Sn the number
Sn= |
1
Pn(-1)
|
|
ó õ
|
1
0
|
|
Pn(-1)-Pn(x)
1+x
|
w(x)dx. |
|
|
Notice that Sn is a linear combination of the number (ak)0 £ k < n since if we write Pn(x) = åk=0n pk (-x)k, we have
easily
Sn = |
1
Pn(-1)
|
|
n-1 å
k=0
|
ck (-1)k ak with ck= |
n å
j = k+1
|
pj |
| (6) |
The number Sn satisfies
Sn = |
ó õ
|
1
0
|
|
w(x)
1+x
|
dx - |
ó õ
|
1
0
|
|
Pn(x)w(x)
Pn(-1)(1+x)
|
dx = S- |
ó õ
|
1
0
|
|
Pn(x)w(x)
Pn(-1)(1+x)
|
dx. |
|
|
Therefore we deduce
| Sn-S| £ |
1
| Pn(-1)|
|
|
ó õ
|
1
0
|
|
| Pn(x)| w(x)
(1+x)
|
dx £ |
Mn
|Pn(-1)|
|
| S| , Mn= |
sup
x Î [0,1]
|
|pn(x)| |
|
|
This inequality suggests to choose polynomials with small value of Mn/|Pn(-1)|.
6.2.1 Choice of a family of polynomials
- A first possible choice is
Pn(x) = (1-x)n = |
n å
k = 0
|
|
æ ç
è
|
n
k
|
ö ÷
ø
|
(-x)k, for which Pn(-1) = 2n,Mn = 1. |
|
It leads to the acceleration
where
Sn = |
1 2n
|
|
n-1 å
k = 0
|
(-1)kckak, ck = |
n å
j = k+1
|
|
æ ç
è
|
n
j
|
ö ÷
ø
|
. |
|
This choice corresponds in fact to Euler's method.
- Another choice is
Pn(x) = (1-2x)n = |
n å
k = 0
|
2k |
æ ç
è
|
n
k
|
ö ÷
ø
|
(-x)k, for which Pn(-1) = 3n,Mn = 1. |
|
It leads to the acceleration
where
Sn = |
1 3n
|
|
n-1 å
k = 0
|
(-1)kckak, ck = |
n å
j = k+1
|
2j |
æ ç
è
|
n
j
|
ö ÷
ø
|
. |
|
- Chebyshev's polynomials (see [1]) shifted to [0,1]
provide a more efficient acceleration. They satisfy the relation
and explicitly writes in the form
Pn(x) = |
n å
j = 0
|
|
n n+j
|
|
æ ç
è
|
n+j
2j
|
ö ÷
ø
|
4j(-x)j. |
|
The relation (7) show that Mn=1 and |Pn(-1)| ³ [ 1/2](3+Ö8)n > 5.828n/2. This family
leads to the following acceleration process :
where
Sn = |
1 Pn(-1)
|
|
n-1 å
k = 0
|
(-1)kckak, ck = |
n å
j = k+1
|
|
n n+j
|
|
æ ç
è
|
n+j
2j
|
ö ÷
ø
|
4j. |
|
- Other families of orthogonal polynomials such as Legendre's
polynomials, Niven's polynomial may give interesting accelerations. More
details and results may be found in the very interesting paper [4].
Once the choice of a sequence of polynomials is made, it can be applied to
compute the value of many alternating series such as
|
|
| |
|
ak= |
1
2k+1
|
= |
ó õ
|
1
0
|
xk |
x-1/2
2
|
dx |
|
(1-2-s)z(s)= |
¥ å
k=0
|
|
(-1)k
(k+1)s
|
, |
|
|
ak= |
1
(k+1)s
|
= |
1
G(s)
|
|
ó õ
|
1
0
|
xk | log(x)| s-1dx. |
|
|
Notice that this latest method is very efficient and may be used
to compute the value of the zeta function at values of s with
Â(s) > 0 (see [3]). Another beautiful alternating
series whose convergence can be accelerated in this way is
log(2) |
æ è
|
g- |
1
2
|
log(2) |
ö ø
|
= |
log(2)
2
|
- |
log(3)
3
|
+ |
log(4)
4
|
- |
log(5)
5
|
+¼ |
|
where g is the Euler constant (see [6] for a proof of
this formula).
7 Acceleration with the Zeta function
If you know how to evaluate the Zeta function at integers values, there is
an easy and convenient way to transform your original series in a geometric
converging one (based on [5]). Suppose that you want to estimate
a series which has the form
where A is a constant and where the analytic function f may be written
then
S=A+ |
¥ å
k=2
|
|
æ è
|
¥ å
n=2
|
|
an
kn
|
ö ø
|
=A+ |
¥ å
n=2
|
an |
æ è
|
¥ å
k=2
|
|
1
kn
|
ö ø
|
|
|
|
hence (8) may be transformed to
S=A+ |
¥ å
n=2
|
an( z(n)-1) . |
| (9) |
Observe that
therefore the transformed series (9) has a geometric rate
of convergence. This rate may be improved if a few terms of the original
series are computed, this time the limit is given by
S=A+ |
M å
k=2
|
f |
æ è
|
1
k
|
ö ø
|
+ |
¥ å
k=M+1
|
f |
æ è
|
1
k
|
ö ø
|
=A+ |
M å
k=2
|
f |
æ è
|
1
k
|
ö ø
|
+ |
¥ å
n=2
|
an |
æ è
|
¥ å
k=M+1
|
|
1
kn
|
ö ø
|
|
|
|
and again
S=A+f |
æ è
|
1
2
|
ö ø
|
+...+f |
æ è
|
1
M
|
ö ø
|
+ |
¥ å
n=2
|
an |
æ è
|
z(n)-1- |
1
2n
|
-...- |
1
Mn
|
ö ø
|
|
|
|
but this time
z(n)-1- |
1
2n
|
-...- |
1
Mn
|
=z(n,M+1) ~ |
1
(M+1)n
|
|
|
|
and z(s,a) is the Hurwitz Zeta Function. The rate of
convergence remains geometric but this rate may be taken as large as desired
by taking a large enough value for M.
7.1 Examples
- The first natural example comes with the (almost) definition series
for Euler's constant
S=g = 1+ |
¥ å
k=2
|
|
æ è
|
1
k
|
+log |
æ è
|
1- |
1
k
|
ö ø
|
ö ø
|
|
|
|
here
f(z)=z+log(1-z)=- |
¥ å
n=2
|
|
zn
n
|
|
|
|
and the transformed series is
g = 1- |
¥ å
n=2
|
|
( z(n)-1)
n
|
. |
|
|
- Another interesting series is Mercator's relation :
S=log(2)=1- |
1
2
|
+ |
1
3
|
- |
1
4
|
+...= |
1
2
|
+ |
¥ å
k=2
|
|
1
(2k-1)2k
|
|
|
|
and this time
f(z)= |
z2
2(2-z)
|
= |
¥ å
n=2
|
|
zn
2n
|
|
|
|
giving
log(2)= |
1
2
|
+ |
¥ å
n=2
|
|
( z(n)-1)
2n
|
|
|
|
and if we compute two more terms
log(2)= |
37
60
|
+ |
¥ å
n=2
|
|
(z(n)-1-1/2n-1/3n)
2n
|
. |
|
|
- A check relation
Observe that if we take for the function f ,an=1 for every n
f(z)= |
¥ å
n=2
|
zn= |
1
1-z
|
-1-z= |
z2
1-z
|
|
|
|
so that for k > 1
f |
æ è
|
1
k
|
ö ø
|
= |
1
k(k-1)
|
= |
1
k-1
|
- |
1
k
|
|
|
|
and because clearly
S= |
¥ å
k=2
|
f |
æ è
|
1
k
|
ö ø
|
= |
¥ å
k=2
|
|
æ è
|
1
k-1
|
- |
1
k
|
ö ø
|
=1, |
|
|
we find the relation
With the same method come the two other relations
which may be used to check the evaluations of the Zeta function on
consecutive integer values.
References
- [1]
- M. Abramowitz and I. Stegun, Handbook of
Mathematical Functions, Dover, New York, (1964)
- [2]
- A.C. Aitken, On Bernoulli's numerical solution of
algebraic equations, Proc. Roy. Soc. Edinburgh, (1926), vol. 46, p. 289-305
- [3]
- P. Borwein, An efficient algorithm for the Riemann
Zeta function, (1995)
- [4]
- H. Cohen, F. Rodriguez Villegas, D. Zagier, Convergence acceleration of alternating series, Bonn, (1991)
- [5]
- P. Flajolet and I. Vardi, Zeta Function Expansions
of Classical Constants, (1996)
- [6]
- X. Gourdon and P. Sebah, Numbers, Constants and
Computation, World Wide Web site at the adress http://numbers.computation.free.fr/Constants/constants.html, (1999)
- [7]
- R.L. Graham, D.E. Knuth and O. Patashnik, Concrete
Mathematics, Addison-Wesley, (1994)
- [8]
- K. Knopp, Theory and application of infinite series,
Blackie & Son, London, (1951)
- [9]
- C. Maclaurin, A Treatise of fluxions, Edinburgh,
(1742)
- [10]
- L.W. Richardson, The deferred Approach to the
Limit, Philosophical Transactions of the Royal Society of London, (1927),
serie A, vol. 226
- [11]
- P. Wynn, On a device for computing the em(Sn) transformation, MTAC, (1956), vol. 10, p. 91-96
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