# Collection of series for p

## 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.

 p 4
 =
 1- 1 3 + 1 5 - 1 7 +...       (Leibniz-Gregory-Madhava)
 p2 16
 =
 ¥åk = 0 (-1)k k + 1 æè 1+ 1 3 +...+ 1 2k + 1 öø (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
 =
 ¥åk = 1 3k - 1 4k z(k + 1)       (Flajolet-Vardi)
 p 4
 =
 ¥åk = 1 arctan æè 1 k2 + k + 1 öø (Knopp)
 1 p
 =
 ¥åk = 1 1 2k + 1 tan æè p 2k + 1 öø (Euler)
 pÖ2 4
 =
 1+ 1 3 - 1 5 - 1 7 + 1 9 + 1 11 -...
 pÖ3 6
 =
 1- 1 5 + 1 7 - 1 11 + 1 13 - 1 17 +...
 pÖ3 9
 =
 1- 1 2 + 1 4 - 1 5 + 1 7 - 1 8 +...
 pÖ3 6
 =
 ¥åk = 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
 =
 ¥åk = 1 1 k2 =1+ 1 22 + 1 32 +...
 p4 90
 =
 ¥åk = 1 1 k4 =1+ 1 24 + 1 34 +...
 p6 945
 =
 ¥åk = 1 1 k6 =1+ 1 26 + 1 36 +...
 4p| B2p| p2p 2(2p)!
 =
 ¥åk = 1 1 k2p = z(2p)

### 3.2  Odd integers

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

### 3.3  All integers alternating

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

### 3.4  Odd integers alternating

 p3 32
 =
 ¥åk = 0 (-1)k (2k + 1)3 =1- 1 33 + 1 53 -...
 5p5 1536
 =
 ¥åk = 0 (-1)k (2k + 1)5 =1- 1 35 + 1 55 -...
 61p7 184320
 =
 ¥åk = 0 (-1)k (2k + 1)7 =1- 1 37 + 1 57 -...
 | E2p| p2p + 1 4p + 1(2p)!
 =
 ¥åk = 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 æè 1 p öø = åk ³ 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(Ö3).

### 4.1  Two terms formulae

 p 2
 =
 2L(Ö2) + L(2Ö2)       (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(3Ö3) + L(4Ö3)
 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)       (Störmer)
 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)       (Störmer)
 p 4
 =
 88L(172) + 51L(239) + 32L(682) + 44L(5357) + 68L(12943)      (Störmer)
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 = ¥åk = 0 æè 4 8k + 1 - 2 8k + 4 - 1 8k + 5 - 1 8k + 6 öø 1 16k .

Other such formulae are available:

 p
 =
 ¥åk = 0 æè 2 4k + 1 + 2 4k + 2 + 1 4k + 3 öø (-1)k 4k ,
 p
 =
 ¥åk = 0 æè 2 8k+1 + 1 4k+1 + 1 8k+3 - 1 16k+10 - 1 16k+12 - 1 32k+28 öø 1 16k ,
 p
 =
 1 64 ¥åk = 0 æè - 32 4k+1 - 1 4k+3 + 256 10k+1 - 64 10k+3 - 4 10k+5 - 4 10k+7 + 1 10k+9 öø (-1)k 1024k ,
 pÖ2
 =
 ¥åk = 0 æè 4 6k+1 + 1 6k+3 + 1 6k+5 öø (-1)k 8k ,
 8p2 9
 =
 ¥åk = 0 æè 16 (6k+1)2 - 24 (6k+2)2 - 8 (6k+3)2 - 6 (6k+4)2 + 1 (6k+5)2 öø 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 æè 1 2 öø 3 +9 æè 1.3 2.4 öø 3 -13 æè 1.3.5 2.4.6 öø 3 +...
 4 p
 =
 1+ æè 1 2 öø 2 + æè 1 2.4 öø 2 + æè 1.3 2.4.6 öø 2 + æè 1.3.5 2.4.6.8 öø 2 +...       (Forsyth)
 1 p
 =
 ¥åk = 0 (2k)!3 (k!)6 (42k + 5) 212k + 4
 1 p
 =
 1 72 ¥åk = 0 (-1)k (4k)! (k!)444k (23 + 260k) 182k
 1 p
 =
 1 3528 ¥åk = 0 (-1)k (4k)! (k!)444k (1123 + 21460k) 8822k
 1 p
 =
 2Ö2 9801 ¥åk = 0 (4k)! (k!)444k (1103 + 26390k) 994k
 1 p
 =
 12 ¥åk = 0 (-1)k (6k)! (3k)!(k!)3 (13591409 + 545140134k) 6403203k + 3/2 (Chudnovsky)
 1 p
 =
 12 ¥åk = 0 (-1)k (6k)! (3k)!(k!)3 (A + Bk) C3k + 3/2 (Borwein)
In the last formula

 A
 =
 1657145277365+212175710912 Ö 61 ,
 B
 =
 107578229802750+13773980892672 Ö 61 ,
 C
 =
 5280(236674+30303 Ö 61 ),

## 7  Other series

 p - 3 6
 =
 ¥åk = 1 (-1)k + 1 36k2 - 1
 p - 3
 =
 ¥åk = 1 (-1)k + 1 k(k + 1)(2k + 1)
 p3 + 8p - 56 8
 =
 ¥åk = 1 (-1)k + 1 k(k + 1)(2k + 1)3
 p 16
 =
 ¥åk odd (-1)(k - 1)/2 k(k4 + 4) (Glaisher)
 p 4
 =
 1 - 16 ¥åk = 0 1 (4k + 1)2(4k + 3)2(4k + 5)2 (Lucas)
 10 - p2
 =
 ¥åk = 1 1 k3(k + 1)3
 p2 - 8 16
 =
 ¥åk = 1 1 (4k2 - 1)2 (Euler)
 32 - 3p2 64
 =
 ¥åk = 1 1 (4k2 - 1)3 (Euler)
 p4 + 30p2 - 384 768
 =
 ¥åk = 1 1 (4k2 - 1)4 (Euler)
 2pÖ3 9
 =
 ¥åk = 0 k!2 (2k + 1)! =1+ 1 6 + 1 30 + 1 140 + 1 630 +...
 2pÖ3 27 + 4 3
 =
 ¥åk = 0 k!2 (2k)! =1+ 1 2 + 1 6 + 1 20 + 1 70 + 1 252 +...
 pÖ3 9
 =
 ¥åk = 1 k!2 (2k)!k
 p2 18
 =
 ¥åk = 1 k!2 (2k)!k2 (Euler)
 17p4 3240
 =
 ¥åk = 1 k!2 (2k)!k4 (Comtet)
 p 3
 =
 ¥åk = 0 (2k)! (2k + 1)16kk!2
 p + 3
 =
 ¥åk = 1 k!2k2k (2k)!
 p
 =
 ¥åk = 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 à l'analyse infinitésimale (french traduction by Labey), Barrois, ainé, 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. Störmer, Sur l'application de la théorie des nombres entiers complexes à la solution en nombres rationnels x1, x2, ..., xn, c1, c2, ..., cn, k de l'équation c1arctg x1+c2 arctg x2+...+cn arctg xn=kp/4, Archiv for Mathematik og Naturvidenskab, (1896), vol. 19