Collection of for $\ $ Page 1 of 9 Collection of series for πππ

(Click here for a Postscript version of this page.) 1 Introduction

There are a great many numbers of series involving the constant π, we provide a selection. The great Swiss mathematician Leonhard Euler (1707-1783) discovered many of those. 2 Around Leibniz-Gregory-Madhava series

π 1 1 1 = 1− + − +... (Leibniz −Gregory −Madhava) 4 3 5 7 ∞ 2 k  1 1  π = ( −1) 1+ +...+ (Knopp) 16 ∑ k + 1  3 2k + 1  k = 0 π 3 1 1 1 = + − + −... (Nilakantha) 4 4 2.3.4 4.5.6 6.7.8 π 1 1.2 1.2.3 = 1+ + + +... (Euler) 2 3 3.5 3.5.7 π 1.2 1.2.3 1.2.3.4 = + + +... 2 1.3 1.3.5 1.3.5.7 ∞ 3k − 1 π = ∑ k ζ(k + 1) (Flajolet −Vardi) 4 k = 1 ∞ π  1  = ∑ arctan 2 (Knopp) 4  k + k + 1  k = 1 ∞ 1 1  π  = ∑ k + 1 tan k + 1 (Euler) π 2  2  k = 1 π√2 1 1 1 1 1 = 1+ − − + + −... 4 3 5 7 9 11 π√3 1 1 1 1 1 = 1− + − + − +... 6 5 7 11 13 17 π√3 1 1 1 1 1 = 1− + − + − +... 9 2 4 5 7 8 ∞ π√3 (−1) k = ∑ k (Sharp) 6 3 (2 k + 1) k = 0

3 Euler's series http://numbers.computation.free.fr/Constants/Pi/piSeries.html 03/12/2006 Collection of series for $\pi $ Page 2 of 9 It was a great problem to find the limit of the series

1 1 1 1+ + +...+ 2 +..., 4 9 k 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

∞ π2 1 1 1 = ∑ 2 =1+ 2 + 2 +... 6 k 2 3 k = 1 ∞ π4 1 1 1 = ∑ 4 =1+ 4 + 4 +... 90 k 2 3 k = 1 ∞ π6 1 1 1 = ∑ 6 =1+ 6 + 6 +... 945 k 2 3 k = 1 ∞ p 2p 4 | B 2p| π 1 = ∑ 2p = ζ (2 p) 2(2 p)! k k = 1

3.2 Odd integers

∞ π2 1 1 1 = ∑ 2 =1+ 2 + 2 +... 8 (2 k + 1) 3 5 k = 0 ∞ π4 1 1 1 = ∑ 4 =1+ 4 + 4 +... 96 (2 k + 1) 3 5 k = 0 ∞ π6 1 1 1 = ∑ 6 =1+ 6 + 6 +... 960 (2 k + 1) 3 5 k = 0 ∞ p 2p (4 − 1)| B 2p| π 1 = ∑ 2p 2(2 p)! (2 k + 1) k = 0

3.3 All integers alternating

http://numbers.computation.free.fr/Constants/Pi/piSeries.html 03/12/2006 Collection of series for $\pi $ Page 3 of 9

∞ π2 (−1) k + 1 1 1 = ∑ 2 =1 − 2 + 2 −... 12 k 2 3 k = 1 ∞ 7π4 (−1) k + 1 1 1 = ∑ 4 =1 − 4 + 4 −... 720 k 2 3 k = 1 ∞ 31 π6 (−1) k + 1 1 1 = ∑ 6 =1 − 6 + 6 −... 30240 k 2 3 k = 1 ∞ p 2p k + 1 (4 − 2)| B 2p| π ( −1) = ∑ 2p 2(2 p)! k k = 1

3.4 Odd integers alternating

∞ π3 (−1) k 1 1 = ∑ 3 =1 − 3 + 3 −... 32 (2 k + 1) 3 5 k = 0 ∞ 5π5 (−1) k 1 1 = ∑ 5 =1 − 5 + 5 −... 1536 (2 k + 1) 3 5 k = 0 ∞ 61 π7 (−1) k 1 1 = ∑ 7 =1 − 7 + 7 −... 184320 (2 k + 1) 3 5 k = 0 ∞ 2p + 1 | E | π (−1) k 2p = p + 1 ∑ 2p + 1 4 (2 p)! (2 k + 1) k = 0

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

1 1 1 1 1 5 B = 1, B = − , B = , B = − , B = , B = − , B = ,... 0 1 2 2 6 4 30 6 42 8 30 10 66

E0 = 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.

http://numbers.computation.free.fr/Constants/Pi/piSeries.html 03/12/2006 Collection of series for $\pi $ Page 4 of 9

π2 1 1 1 1 1 1 = 2 + 2 + 2 + 2 + 2 + 2 +... 20 2 3 5 7 8 11 π4 1 1 1 1 1 1 = 4 + 4 + 4 + 4 + 4 + 4 +... 1260 2 3 5 7 8 11 4π6 1 1 1 1 1 1 = 6 + 6 + 6 + 6 + 6 + 6 +... 225225 2 3 5 7 8 11 ζ2(2 p) − ζ (4 p) 1 1 1 1 1 1 = 2p + 2p + 2p + 2p + 2p + 2p +... 2ζ(2 p) 2 3 5 7 8 11

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

9 1 1 1 1 1 1 2 = 2 + 2 + 2 + 2 + 2 + 2 +... 2π 2 3 5 7 11 13 15 1 1 1 1 1 1 4 = 4 + 4 + 4 + 4 + 4 + 4 +... 2π 2 3 5 7 11 13 11340 1 1 1 1 1 1 6 = 6 + 6 + 6 + 6 + 6 + 6 +... 691 π 2 3 5 7 11 13 ζ2(2 p) − ζ (4 p) 1 1 1 1 1 1 = 2p + 2p + 2p + 2p + 2p + 2p +... 2ζ(2 p)ζ(4 p) 2 3 5 7 11 13

4 Machin's formulae

By mean of the

 1  (−1) k L( p) = arctan =  p  ∑ 2k + 1 (2 k + 1) p k ≥ 0 numerous more or less efficient formulae to express π are available ( [ 1], [ 5], [ 7], [ 9]).

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

4.1 Two terms formulae

π = 2L( √2) + L(2 √2) (Wetherfield) 2 π = L(2) + L(3) (Hutton) 4 π = 2L(3) + L(7) (Hutton) 4 π = 4L(5) − L(239) (Machin) 4 π = 2L(3 √3) + L(4 √3) 6 http://numbers.computation.free.fr/Constants/Pi/piSeries.html 03/12/2006 Collection of series for $\pi $ Page 5 of 9

π = 5L(7) + 2L(79/3) (Euler) 4 π = 5L(278/29) + 7L(79/3) 4

4.2 Three terms and more formulae

π = L(2) + L(5) + L(8) (Strassnitzky) 4 π = 4L(5) − L(70) + L(99) (Euler) 4 π = 5L(7) + 4L(53) + 2L(4443) 4 π = 6L(8) + 2L(57) + L(239) (Störmer) 4 π = 8L(10) − L(239) − 4L(515) (Klingenstierna) 4 π = 12L(18) + 8L(57) − 5L(239) (Gauss) 4 π = 22L(38) + 17L(601/7) + 10L(8149/7) (Sebah) 4 π = 44L(57) + 7L(239) − 12L(682) + 24L(12943) (Störmer) 4 π = 88L(172) + 51L(239) + 32L(682) + 44L(5357) + 68L(12943) (Störmer) 4 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 π in basis 2 (see [ 2])

∞  4 2 1 1  1 π = ∑ − − − k .  8k + 1 8k + 4 8k + 5 8k + 6  16 k = 0

Other such formulae are available:

∞  2 2 1  ( −1) k = + + , π ∑   k 4k + 1 4k + 2 4k + 3 4 k = 0

∞ http://numbers.computation.free.fr/Constants/Pi/piSeries.html 03/12/2006 Collection of series for $\pi $ Page 6 of 9

π =  2 1 1 1 1 1  1 ∑ + + − − − ,  8k+1 4k+1 8k+3 16k+10 16k+12 32k+28  16 k k = 0 ∞ 1  32 1 256 64 4 4 1  ( −1) k = + + , π ∑  − − − − −  k 64 4k+1 4k+3 10k+1 10k+3 10k+5 10k+7 10k+9 1024 k = 0 ∞  4 1 1  ( −1) k π√2 = + + , ∑  6k+1 6k+3 6k+5  k 8 k = 0 ∞ 8 π2  16 24 8 6 1  1 = ∑ 2 − 2 − 2 − 2 + 2 k . 9  (6k+1) (6k+2) (6k+3) (6k+4) (6k+5)  64 k = 0

The series with 1024 k 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]).

3 3 3 2  1   1.3   1.3.5  = 1−5 +9 −13 +... π  2   2.4   2.4.6  2 2 2 2 4  1   1   1.3   1.3.5  = 1+ + + + +... (Forsyth) π  2   2.4   2.4.6   2.4.6.8  ∞ 1 (2k)! 3 (42k + 5) = ∑ 6 12k + 4 π (k!) 2 k = 0 ∞ 1 1 k (4k)! (23 + 260k) = ∑ (−1) 4 4k 2k π 72 (k!) 4 18 k = 0 ∞ 1 1 k (4k)! (1123 + 21460k) = ∑ (−1) 4 4k 2k π 3528 (k!) 4 882 k = 0 ∞ 1 2√2 (4k)! (1103 + 26390k) = ∑ 4 4k 4k π 9801 (k!) 4 99 k = 0 ∞ 1 k (6k)! (13591409 + 545140134k) = 12 ∑ (−1) 3 3k + 3/2 (Chudnovsky) π (3k)!(k!) 640320 k = 0

∞ http://numbers.computation.free.fr/Constants/Pi/piSeries.html 03/12/2006 Collection of series for $\pi $ Page 7 of 9

1 = (6k)! (A + Bk) π 12 ∑ (−1) k (Borwein) (3k)!(k!) 3 C3k + 3/2 k = 0 In the last formula

61 A = 1657145277365+212175710912 √ ,

61 B = 107578229802750+13773980892672 √ ,

61 C = 5280(236674+30303 √ ), and each additional term in the series adds about 31 digits ... 7 Other series

∞ π − 3 ( −1) k + 1 = ∑ 2 6 36k − 1 k = 1 ∞ ( −1) k + 1 π − 3 = ∑ k(k + 1)(2k + 1) k = 1 ∞ π3 + 8 π − 56 ( −1) k + 1 = ∑ 3 8 k(k + 1)(2k + 1) k = 1 ∞ π ( −1) (k − 1)/2 = (Glaisher) ∑ 4 16 k(k + 4) k odd ∞ π 1 = 1 − 16 ∑ 2 2 2 (Lucas) 4 (4k + 1) (4k + 3) (4k + 5) k = 0 ∞ 2 1 10 − π = ∑ 3 3 k (k + 1) k = 1 ∞ π2 − 8 1 = ∑ 2 2 (Euler) 16 (4k − 1) k = 1

∞ http://numbers.computation.free.fr/Constants/Pi/piSeries.html 03/12/2006 Collection of series for $\pi $ Page 8 of 9

2 = 32 − 3π 1 64 ∑ (Euler) (4k 2 − 1) 3 k = 1 ∞ π4 + 30 π2 − 384 1 = ∑ 2 4 (Euler) 768 (4k − 1) k = 1 ∞ 2 π√3 2 1 1 1 1 = k! =1+ + + + +... 9 ∑ (2k + 1)! 6 30 140 630 k = 0 ∞ 2π√3 4 2 1 1 1 1 1 + = k! =1+ + + + + +... 27 3 ∑ (2k)! 2 6 20 70 252 k = 0 ∞ π√3 2 = k! 9 ∑ (2k)!k k = 1 ∞ π2 k! 2 = ∑ 2 (Euler) 18 (2k)!k k = 1 ∞ 17 π4 k! 2 = ∑ 4 (Comtet) 3240 (2k)!k k = 1 ∞ π (2k)! = ∑ k 2 3 (2k + 1)16 k! k = 0 ∞ k! 2k2 k π + 3 = ∑ (2k)! k = 1 ∞ (25k − 3)k!(2k)! π = ∑ k − 1 (Gosper 1974) 2 (3k)! k = 0

References

[1] J. Arndt and C. Haenel, π− 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 http://numbers.computation.free.fr/Constants/Pi/piSeries.html 03/12/2006 Collection of series for $\pi $ Page 9 of 9 [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 π, The American Mathematical Monthly, (1938), pp. 657-664

[8] S. Ramanujan, Modular equations and approximations to π, 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, ..., x n, c 1, c 2, ..., c n, k de l'équation c1arctg x 1+c 2 arctg x 2+...+c n arctg x n=k π/4, Archiv for Mathematik og Naturvidenskab, (1896), vol. 19

http://numbers.computation.free.fr/Constants/Pi/piSeries.html 03/12/2006