Collection of series for p

(Click here for a Postscript version of this page.)

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 ) p 2 16

= ¥

å

k = 0

( - 1)k k + 1

æ

è 1+ 1 3

+...+ 1 2k + 1

ö

ø ( 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 = ¥

å

k = 1

3k - 1 4k

z (k + 1) ( Flajolet - Vardi ) p 4

= ¥

å

k = 1

arctan æ

è 1 k2 + k + 1

ö

ø ( Knopp ) 1 p

= ¥

å

k = 1

1 2k + 1

tan æ

è p 2k + 1

ö

ø ( Euler ) p Ö 2 4

= 1+ 1 3

- 1 5

- 1 7

+ 1 9

+ 1 11

- ... p Ö 3 6

= 1 - 1 5

+ 1 7

- 1 11

+ 1 13

- 1 17

+... p Ö 3 9

= 1 - 1 2

+ 1 4

- 1 5

+ 1 7

- 1 8

+... p Ö 3 6

= ¥

å

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

+...,

3.1 All integers





p 2 6

= ¥

å

k = 1

1 k2

=1+ 1 22

+ 1 32

+... p 4 90

= ¥

å

k = 1

1 k4

=1+ 1 24

+ 1 34

+... p 6 945

= ¥

å

k = 1

1 k6

=1+ 1 26

+ 1 36

+... 4p| B 2p | p 2p 2(2p)!

= ¥

å

k = 1

1 k2p

= z (2p)

3.2 Odd integers





p 2 8

= ¥

å

k = 0

1 (2k + 1)2

=1+ 1 32

+ 1 52

+... p 4 96

= ¥

å

k = 0

1 (2k + 1)4

=1+ 1 34

+ 1 54

+... p 6 960

= ¥

å

k = 0

1 (2k + 1)6

=1+ 1 36

+ 1 56

+... (4p - 1)| B 2p | p 2p 2(2p)!

= ¥

å

k = 0

1 (2k + 1)2p



3.3 All integers alternating





p 2 12

= ¥

å

k = 1

( - 1)k + 1 k2

=1 - 1 22

+ 1 32

- ... 7 p 4 720

= ¥

å

k = 1

( - 1)k + 1 k4

=1 - 1 24

+ 1 34

- ... 31 p 6 30240

= ¥

å

k = 1

( - 1)k + 1 k6

=1 - 1 26

+ 1 36

- ... (4p - 2)| B 2p | p 2p 2(2p)!

= ¥

å

k = 1

( - 1)k + 1 k2p



3.4 Odd integers alternating





p 3 32

= ¥

å

k = 0

( - 1)k (2k + 1)3

=1 - 1 33

+ 1 53

- ... 5 p 5 1536

= ¥

å

k = 0

( - 1)k (2k + 1)5

=1 - 1 35

+ 1 55

- ... 61 p 7 184320

= ¥

å

k = 0

( - 1)k (2k + 1)7

=1 - 1 37

+ 1 57

- ... | E 2p | p 2p + 1 4p + 1(2p)!

= ¥

å

k = 0

( - 1)k (2k + 1)2p + 1



B n and E n are respectively Bernoulli's numbers and Euler's numbers.





B 0 = 1, B 1 = - 1 2

, B 2 = 1 6

, B 4 = - 1 30

, B 6 = 1 42

, B 8 = - 1 30

, B 10 = 5 66

,... E 0 = 1, E 2 = - 1, E 4 =5, E 6 = - 61, E 8 =1385, E 10 = - 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.





p 2 20

= 1 22

+ 1 32

+ 1 52

+ 1 72

+ 1 82

+ 1 112

+... p 4 1260

= 1 24

+ 1 34

+ 1 54

+ 1 74

+ 1 84

+ 1 114

+... 4 p 6 225225

= 1 26

+ 1 36

+ 1 56

+ 1 76

+ 1 86

+ 1 116

+... z 2(2p) - z (4p) 2 z (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 2 p 2

= 1 22

+ 1 32

+ 1 52

+ 1 72

+ 1 112

+ 1 132

+... 15 2 p 4

= 1 24

+ 1 34

+ 1 54

+ 1 74

+ 1 114

+ 1 134

+... 11340 691 p 6

= 1 26

+ 1 36

+ 1 56

+ 1 76

+ 1 116

+ 1 136

+... z 2(2p) - z (4p) 2 z (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 æ

è 1 p

ö

ø =

å

k ³ 0

( - 1)k (2k + 1)p2k + 1



p

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

4.1 Two terms formulae





p 2

= 2L( Ö 2) + L(2 Ö 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 Ö 3) + L(4 Ö 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 ö 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 ö rmer ) p 4

= 88L(172) + 51L(239) + 32L(682) + 44L(5357) + 68L(12943) ( St ö rmer )

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 = ¥

å

k = 0

æ

è 4 8k + 1

- 2 8k + 4

- 1 8k + 5

- 1 8k + 6

ö

ø 1 16k

.

Other such formulae are available:





p = ¥

å

k = 0

æ

è 2 4k + 1

+ 2 4k + 2

+ 1 4k + 3

ö

ø ( - 1)k 4k

, p = ¥

å

k = 0

æ

è 2 8k+1

+ 1 4k+1

+ 1 8k+3

- 1 16k+10

- 1 16k+12

- 1 32k+28

ö

ø 1 16k

, p = 1 64

¥

å

k = 0

æ

è - 32 4k+1

- 1 4k+3

+ 256 10k+1

- 64 10k+3

- 4 10k+5

- 4 10k+7

+ 1 10k+9

ö

ø ( - 1)k 1024k

, p Ö 2 = ¥

å

k = 0

æ

è 4 6k+1

+ 1 6k+3

+ 1 6k+5

ö

ø ( - 1)k 8k

, 8 p 2 9

= ¥

å

k = 0

æ

è 16 (6k+1)2

- 24 (6k+2)2

- 8 (6k+3)2

- 6 (6k+4)2

+ 1 (6k+5)2

ö

ø 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 æ

è 1 2

ö

ø 3





+9 æ

è 1.3 2.4

ö

ø 3





- 13 æ

è 1.3.5 2.4.6

ö

ø 3





+... 4 p

= 1+ æ

è 1 2

ö

ø 2





+ æ

è 1 2.4

ö

ø 2





+ æ

è 1.3 2.4.6

ö

ø 2





+ æ

è 1.3.5 2.4.6.8

ö

ø 2





+... ( Forsyth ) 1 p

= ¥

å

k = 0

(2k)!3 (k!)6

(42k + 5) 212k + 4

1 p

= 1 72

¥

å

k = 0

( - 1)k (4k)! (k!)444k

(23 + 260k) 182k

1 p

= 1 3528

¥

å

k = 0

( - 1)k (4k)! (k!)444k

(1123 + 21460k) 8822k

1 p

= 2 Ö 2 9801

¥

å

k = 0

(4k)! (k!)444k

(1103 + 26390k) 994k

1 p

= 12 ¥

å

k = 0

( - 1)k (6k)! (3k)!(k!)3

(13591409 + 545140134k) 6403203k + 3/2

( Chudnovsky ) 1 p

= 12 ¥

å

k = 0

( - 1)k (6k)! (3k)!(k!)3

(A + Bk) C3k + 3/2

( Borwein )





A = 1657145277365+212175710912

Ö

61

, B = 107578229802750+13773980892672

Ö

61

, C = 5280(236674+30303

Ö

61

),

7 Other series





p - 3 6

= ¥

å

k = 1

( - 1)k + 1 36k2 - 1

p - 3 = ¥

å

k = 1

( - 1)k + 1 k(k + 1)(2k + 1)

p 3 + 8 p - 56 8

= ¥

å

k = 1

( - 1)k + 1 k(k + 1)(2k + 1)3

p 16

= ¥

å

k odd

( - 1)(k - 1)/2 k(k4 + 4)

( Glaisher ) p 4

= 1 - 16 ¥

å

k = 0

1 (4k + 1)2(4k + 3)2(4k + 5)2

( Lucas ) 10 - p 2 = ¥

å

k = 1

1 k3(k + 1)3

p 2 - 8 16

= ¥

å

k = 1

1 (4k2 - 1)2

( Euler ) 32 - 3 p 2 64

= ¥

å

k = 1

1 (4k2 - 1)3

( Euler ) p 4 + 30 p 2 - 384 768

= ¥

å

k = 1

1 (4k2 - 1)4

( Euler ) 2 p Ö 3 9

= ¥

å

k = 0

k!2 (2k + 1)!

=1+ 1 6

+ 1 30

+ 1 140

+ 1 630

+... 2 p Ö 3 27

+ 4 3

= ¥

å

k = 0

k!2 (2k)!

=1+ 1 2

+ 1 6

+ 1 20

+ 1 70

+ 1 252

+... p Ö 3 9

= ¥

å

k = 1

k!2 (2k)!k

p 2 18

= ¥

å

k = 1

k!2 (2k)!k2

( Euler ) 17 p 4 3240

= ¥

å

k = 1

k!2 (2k)!k4

( Comtet ) p 3

= ¥

å

k = 0

(2k)! (2k + 1)16kk!2

p + 3 = ¥

å

k = 1

k!2k2k (2k)!

p = ¥

å

k = 0

(25k - 3)k!(2k)! 2k - 1(3k)!

( Gosper 1974 )

References