Collection of series for p

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1��Introduction

There are a great many numbers of series involving the constant p, we provide a selection. The great Swiss mathematician Leonhard Euler (1707-1783) discovered many of those.

2��Around Leibniz-Gregory-Madhava series


p
4
=
1- 1
3
 + 1
5
 - 1
7
 +...������ (Leibniz-Gregory-Madhava)
�p2
16
=
�
�
k = 0�
 �(-1)k
k + 1
 �
� 		1+ �1
3
 +...+ �1
2k + 1
 �
� 		������ (Knopp)
p
4
=
3
4
 + 1
2.3.4
 - 1
4.5.6
 + 1
6.7.8
 -...������ (Nilakantha)
p
2
=
1+ 1
3
 + 1.2
3.5
 + 1.2.3
3.5.7
 +...������ (Euler)
p
2
=
1.2
1.3
 + 1.2.3
1.3.5
 + 1.2.3.4
1.3.5.7
 +...
p
=
�
�
k = 1�
 3k - 1
4k
 z(k + 1)������ (Flajolet-Vardi)
p
4
=
�
�
k = 1�
 arctan �
� 		1
k2 + k + 1
 �
� 		������ (Knopp)
1
p
=
�
�
k = 1�
 1
2k + 1
 tan �
� 		p
2k + 1
 �
� 		������ (Euler)
p�2
4
=
1+ 1
3
 - 1
5
 - 1
7
 + 1
9
 + 1
11
 -...
p�3
6
=
1- 1
5
 + 1
7
 - 1
11
 + 1
13
 - 1
17
 +...
p�3
9
=
1- 1
2
 + 1
4
 - 1
5
 + 1
7
 - 1
8
 +...
p�3
6
=
�
�
k = 0�
 (-1)k
3k(2k + 1)
 ������ (Sharp)

3��Euler's series

It was a great problem to find the limit of the series


1+ 1
4
 + 1
9
 +...+ 1
k2
 +...,
some of the greatest mathematicians of the seventeenth century failed to find this limit. After trying unsuccessfully to solve it, Jakob Bernoulli challenged mathematicians with this problem. It was Euler, in 1735, who found the value of the series and most of the following result are also due to him (see [4]).

3.1��All integers


p2
6
=
�
�
k = 1�
 1
k2
 =1+ 1
22
 + 1
32
 +...
p4
90
=
�
�
k = 1�
 1
k4
 =1+ 1
24
 + 1
34
 +...
p6
945
=
�
�
k = 1�
 1
k6
 =1+ 1
26
 + 1
36
 +...
4p| B2p| p2p
2(2p)!
=
�
�
k = 1�
 1
k2p
 �= z(2p)

3.2��Odd integers


�p2
8
=
�
�
k = 0�
 �1
(2k + 1)2
 =1+ 1
32
 + 1
52
 +...
�p4
96
=
�
�
k = 0�
 �1
(2k + 1)4
 =1+ 1
34
 + 1
54
 +...
�p6
960
=
�
�
k = 0�
 �1
(2k + 1)6
 =1+ 1
36
 + 1
56
 +...
�(4p - 1)| B2p| p2p
2(2p)!
=
�
�
k = 0�
 �1
(2k + 1)2p

3.3��All integers alternating


p2
12
=
�
�
k = 1�
 (-1)k + 1
k2
 =1- 1
22
 + 1
32
 -...
7p4
720
=
�
�
k = 1�
 (-1)k + 1
k4
 =1- 1
24
 + 1
34
 -...
31p6
30240
=
�
�
k = 1�
 (-1)k + 1
k6
 =1- 1
26
 + 1
36
 -...
�(4p - 2)| B2p| p2p
2(2p)!
=
�
�
k = 1�
 �(-1)k + 1
k2p

3.4��Odd integers alternating


p3
32
=
�
�
k = 0�
 (-1)k
(2k + 1)3
 =1- 1
33
 + 1
53
 -...
5p5
1536
=
�
�
k = 0�
 (-1)k
(2k + 1)5
 =1- 1
35
 + 1
55
 -...
61p7
184320
=
�
�
k = 0�
 (-1)k
(2k + 1)7
 =1- 1
37
 + 1
57
 -...
| E2p| p2p + 1
4p + 1(2p)!
=
�
�
k = 0�
 (-1)k
(2k + 1)2p + 1

Bn and En are respectively Bernoulli's numbers and Euler's numbers.


B0
=
1, B1= - 1
2
 , B2= 1
6
 , B4= - 1
30
 , B6= 1
42
 , B8= - 1
30
 , B10= 5
66
 ,...
E0
=
1, E2= -1, E4=5, E6= -61, E8=1385, E10= -50251,...

3.5��With prime numbers

In the following series, only the denominators with an odd number of prime factors are taken in account. For example 10=2�5 is omitted because it has two prime factors.


p2
20
=
1
22
 + 1
32
 + 1
52
 + 1
72
 + 1
82
 + 1
112
 +...
p4
1260
=
1
24
 + 1
34
 + 1
54
 + 1
74
 + 1
84
 + 1
114
 +...
4p6
225225
=
1
26
 + 1
36
 + 1
56
 + 1
76
 + 1
86
 + 1
116
 +...
z2(2p) - z(4p)
2z(2p)
=
1
22p
 + 1
32p
 + 1
52p
 + 1
72p
 + 1
82p
 + 1
112p
 +...

If this time the prime factors are also supposed to be different:


9
2p2
=
1
22
 + 1
32
 + 1
52
 + 1
72
 + 1
112
 + 1
132
 +...
15
2p4
=
1
24
 + 1
34
 + 1
54
 + 1
74
 + 1
114
 + 1
134
 +...
11340
691p6
=
1
26
 + �1
36
 + 1
56
 + 1
76
 + 1
116
 + 1
136
 +...
�z2(2p) - z(4p)
2z(2p)z(4p)
=
1
22p
 + 1
32p
 + 1
52p
 + 1
72p
 + 1
112p
 + 1
132p
 +...

4��Machin's formulae

By mean of the function

L(p) = arctan �
� 		1
p
 �
� 		=
�
k � 0�
 (-1)k
(2k + 1)p2k + 1
numerous more or less efficient formulae to express p are available ( [1], [5], [7], [9]).

Observe that the Leibniz-Gregory-Madhava series may be written as p/4 = L(1) and Sharp's series is just p/6 = L(�3).

4.1��Two terms formulae


p
2
=
2L(�2) + L(2�2)������ (Wetherfield)
p
4
=
L(2) + L(3)������ (Hutton)
p
4
=
2L(3) + L(7)������ (Hutton)
p
4
=
4L(5) - L(239)������ (Machin)
p
6
=
2L(3�3) + L(4�3)
p
4
=
5L(7) + 2L(79/3)������ (Euler)
p
4
=
5L(278/29) + 7L(79/3)

4.2��Three terms and more formulae


p
4
=
L(2) + L(5) + L(8)������ (Strassnitzky)
p
4
=
4L(5) - L(70) + L(99)������ (Euler)
p
4
=
5L(7) + 4L(53) + 2L(4443)
p
4
=
6L(8) + 2L(57) + L(239)������ (St�rmer)
p
4
=
8L(10) - L(239) - 4L(515)������ (Klingenstierna)
p
4
=
12L(18) + 8L(57) - 5L(239)������ (Gauss)
p
4
=
22L(38) + 17L(601/7) + 10L(8149/7)������ (Sebah)
p
4
=
44L(57) + 7L(239) - 12L(682) + 24L(12943)������ (St�rmer)
p
4
=
88L(172) + 51L(239) + 32L(682) + 44L(5357) + 68L(12943)������(St�rmer)
For example, more than 100 three terms formulae are known and are easy to generate by mean of dedicated algorithms.

5��BBP series

In 1995, Bailey, Borwein and Plouffe (BBP) found a new kind of formula which allows to compute directly the d-th digit of p in basis 2 (see [2])


p = �
�
k = 0�
 �
� 		4
8k + 1
 - 2
8k + 4
 - 1
8k + 5
 - 1
8k + 6
 �
� 		1
16k
 .

Other such formulae are available:


p
=
�
�
k = 0�
 �
� 		2
4k + 1
 + 2
4k + 2
 + 1
4k + 3
 �
� 		�(-1)k
4k
 ,
p
=
�
�
k = 0�
 �
� 		�2
8k+1
 + �1
4k+1
 + �1
8k+3
 - �1
16k+10
 - �1
16k+12
 - �1
32k+28
 �
� 		1
16k
 ,
p
=
�1
64
 �
�
k = 0�
 �
� 		- �32
4k+1
 - �1
4k+3
 + �256
10k+1
 - �64
10k+3
 - �4
10k+5
 - �4
10k+7
 + �1
10k+9
 �
� 		�(-1)k
1024k
 ,
p�2
=
�
�
k = 0�
 �
� 		�4
6k+1
 + �1
6k+3
 + �1
6k+5
 �
� 		�(-1)k
8k
 ,
�8p2
9
=
�
�
k = 0�
 �
� 		�16
(6k+1)2
 - �24
(6k+2)2
 - �8
(6k+3)2
 - �6
(6k+4)2
 + �1
(6k+5)2
 �
� 		�1
64k
 .

The series with 1024k is efficient and due to F. Bellard (1997).

6��Ramanujan's series

Most of those series and many others were found by the Indian prodigy Srinivasa Ramanujan (1887-1920) ([3], [8]).


2
p
=
1-5 �
� 		1
2
 �
� 		3

�
 +9 �
� 		1.3
2.4
 �
� 		3

�
 -13 �
� 		1.3.5
2.4.6
 �
� 		3

�
 +...
4
p
=
1+ �
� 		1
2
 �
� 		2

�
 + �
� 		1
2.4
 �
� 		2

�
 + �
� 		1.3
2.4.6
 �
� 		2

�
 + �
� 		1.3.5
2.4.6.8
 �
� 		2

�
 +...������ (Forsyth)
1
p
=
�
�
k = 0�
 (2k)!3
(k!)6
 �(42k + 5)
212k + 4
1
p
=
1
72
 �
�
k = 0�
 (-1)k �(4k)!
(k!)444k
 �(23 + 260k)
182k
1
p
=
1
3528
 �
�
k = 0�
 (-1)k �(4k)!
(k!)444k
 �(1123 + 21460k)
8822k
1
p
=
2�2
9801
 �
�
k = 0�
 �(4k)!
(k!)444k
 �(1103 + 26390k)
994k
1
p
=
12 �
�
k = 0�
 (-1)k �(6k)!
(3k)!(k!)3
 �(13591409 + 545140134k)
6403203k + 3/2
 ������ (Chudnovsky)
1
p
=
12 �
�
k = 0�
 (-1)k �(6k)!
(3k)!(k!)3
 �(A + Bk)
C3k + 3/2
 ������ (Borwein)
In the last formula


A
=
1657145277365+212175710912
�
�
61
�
 ,
B
=
107578229802750+13773980892672
�
�
61
�
 ,
C
=
5280(236674+30303
�
�
61
�
 ),
and each additional term in the series adds about 31 digits ...

7��Other series


p - 3
6
=
�
�
k = 1�
 �(-1)k + 1
36k2 - 1
p - 3
=
�
�
k = 1�
 �(-1)k + 1
k(k + 1)(2k + 1)
�p3 + 8p - 56
8
=
�
�
k = 1�
 �(-1)k + 1
k(k + 1)(2k + 1)3
p
16
=
�
�
k odd�
 �(-1)(k - 1)/2
k(k4 + 4)
 ������ (Glaisher)
p
4
=
1 - 16 �
�
k = 0�
 �1
(4k + 1)2(4k + 3)2(4k + 5)2
 ������ (Lucas)
10 - p2
=
�
�
k = 1�
 1
k3(k + 1)3
�p2 - 8
16
=
�
�
k = 1�
 1
(4k2 - 1)2
 ������(Euler)
�32 - 3p2
64
=
�
�
k = 1�
 1
(4k2 - 1)3
 ������ (Euler)
�p4 + 30p2 - 384
768
=
�
�
k = 1�
 1
(4k2 - 1)4
 ������ (Euler)
�2p�3
9
=
�
�
k = 0�
 �k!2
(2k + 1)!
 =1+ �1
6
 + �1
30
 + �1
140
 + �1
630
 +...
2p�3
27
 �+�� 4
3
=
�
�
k = 0�
 k!2
(2k)!
 =1+ �1
2
 + �1
6
 + �1
20
 + �1
70
 + �1
252
 +...
�p�3
9
=
�
�
k = 1�
 k!2
(2k)!k
�p2
18
=
�
�
k = 1�
 k!2
(2k)!k2
 ������(Euler)
�17p4
3240
=
�
�
k = 1�
 �k!2
(2k)!k4
 ������ (Comtet)
p
3
=
�
�
k = 0�
 (2k)!
(2k + 1)16kk!2
p + 3
=
�
�
k = 1�
 �k!2k2k
(2k)!
p
=
�
�
k = 0�
 �(25k - 3)k!(2k)!
2k - 1(3k)!
 ������ (Gosper 1974)

References

[1]
J. Arndt and C. Haenel, p-�Unleashed, Springer, (2001)

[2]
D.H. Bailey, P.B. Borwein and S. Plouffe, On the Rapid Computation of Various Polylogarithmic Constants, Mathematics of Computation, (1997), vol. 66, pp. 903-913

[3]
J.M. Borwein and P.B. Borwein, Ramanujan and Pi, Scientific American, (1988), pp. 112-117

[4]
L. Euler, Introduction � l'analyse infinit�simale (french traduction by Labey), Barrois, ain�, Librairie, (original 1748, traduction 1796), vol. 1

[5]
C.L. Hwang, More Machin-Type Identities, Math. Gaz., (1997), pp. 120-121

[6]
K. Knopp, Theory and application of infinite series, Blackie & Son, London, (1951)

[7]
D.H. Lehmer,�On Arctangent Relations for p, The American Mathematical Monthly, (1938), pp. 657-664

[8]
S. Ramanujan, Modular equations and approximations to p, Quart. J. Pure Appl. Math., (1914), vol. 45, pp. 350-372

[9]
C. St�rmer, Sur l'application de la th�orie des nombres entiers complexes � la solution en nombres rationnels x1, x2, ..., xn, c1, c2, ..., cn, k de l'�quation c1arctg x1+c2 arctg x2+...+cn arctg xn=kp/4, Archiv for Mathematik og Naturvidenskab, (1896), vol. 19