The constant The constant p

3  Ramanujan type formulae

In 1914, the extraordinary mathematician from India, Srinivasa Ramanujan published a set of 14 new formulae ([1]), one of them is famous:

1
p
= 22
9801


k = 0 
(4k)!
(k!)444k
(1103+26390k)
994k
     (*)
The convergence is geometric but very impressive, let's take a look to the first iterates:

x1
=
3.141592(7300133056603139961890252155185995816071...)
x2
=
3.141592653589793(8779989058263060130942166450293...)
x3
=
3.14159265358979323846264(90657027588981566774804...)
x4
=
3.1415926535897932384626433832795(552731599742104...)
x5
=
3.141592653589793238462643383279502884197(6638181...)

Each term adds about 8 digits to the result.

In his article, Ramanujan gave very little details of how he found such formulae. To find those relations, he needed modular equations and hypergeometric functions. It's possible to find a complete demonstration of those series in ([2]). In this book the following theorem is given:

1
p
= gN

k = 0 
(1/4)k(1/2)k(3/4)k
(k!)3
( aN+kbN)gN2k,       N > 2

where (aN,bN,gN) are algebraic numbers whenever N is rational and (a)k is the rising factorial : (a)k = a(a+1)(a+2)...(a+k-1). The expression of (aN,bN,gN) as functions of N is rather complex... For example gN is given by :

gN = 2
gN12+gN-12

where gN is a Ramanujan's invariant.

Ramanujan's formula is given for N = 58, for which g582 = (5+[29])/2 so g58 = 1/992 and a58+kb58 = 22(1103+ 26390k) ([2], [4]).

Other interesting series are found with (N = 6,10,18,22,58), all given by Ramanujan in his article. Here are the value of gN and gN for the corresponding values of N:

g6
=
(1+2)1/6,g6 = 1/3
g10
=
(2+5)1/6,g10 = 1/9
g18
=
(5+26)1/6,g18 = 1/49
g22
=
(1+2)1/2,g22 = 1/99
g58
=
((5+   __
29
 
)/2)1/2,g58 = 1/992 

In the same article, Ramanujan also gave:

1
p
=

k = 0 


2k
k


3

 
(42k+5)
212k+4

This series adds roughly 2 digits per term. A proof was published in [2].

Much later, a few years ago [3], David and Gregory Chudnovsky found a very powerful series of this type :

1
p
= 12

k = 0 
(-1)k (6k)!
(3k)!(k!)3
(13591409+545140134k)
6403203k+3/2

Each term of this series adds about 15 digits...

3.1  Record of computation with the Ramanujan like formulae :

1985 : Gosper found 17,500,000 decimal digits with formula (*).

1994 : The Chudnovsky brothers found 4,044,000,000 digits with their formula.

References

[1]
S. Ramanujan, Modular equations and approximations to p, Quart. J. Pure Appl. Math., (1914), vol. 45, p. 350-372

[2]
J.M. Borwein and P.B. Borwein, Pi and the AGM - A study in Analytic Number Theory and Computational Complexity, A Wiley-Interscience Publication, New York, (1987)

[3]
D. V. Chudnovsky and G. V. Chudnovsky, Approximations and complex multiplication according to Ramanujan, in Ramanujan Revisited, Academic Press Inc., Boston, (1988), p. 375-396 & p. 468-472.

[4]
P. Eymard and J. P. Lafon, Autour du nombre p, Paris, Hermann, (1999)


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