Almkvist Berndt 1988 begins with an investigation of why the agm is such an efficient algorithm, and proves that it converges quadratically. This is an efficient method to calculate 



π


{\displaystyle \pi }

.

With the same notations used in Arithmetic-geometric mean, we can summarize the paper by writing:





π
=



4


a
g
m

(
1
,
1

/



2



)

2




1
−

∑

n
=
1


∞



2

n
+
1


(

a

n


2


−

g

n


2


)





{\displaystyle \pi ={\frac {4\;\mathrm {agm} (1,1/{\sqrt {2}})^{2}}{1-\sum \limits _{n=1}^{\infty }2^{n+1}(a_{n}^{2}-g_{n}^{2})}}}



This allows you to make the approximation, for any large   N:





π
≈



4


a

N


2




1
−

∑

k
=
1


N



2

k
+
1


(

a

k


2


−

g

k


2


)





{\displaystyle \pi \approx {\frac {4\;a_{N}^{2}}{1-\sum \limits _{k=1}^{N}2^{k+1}(a_{k}^{2}-g_{k}^{2})}}}



The purpose of this task is to demonstrate how to use this approximation in order to compute a large number of decimals of 



π


{\displaystyle \pi }

.

pi[n_, prec_] := 
 Module[{a = 1, g = N[1/Sqrt[2], prec], k, s = 0, p = 4}, 
  For[k = 1, k < n, k++,
   {a, g} = {N[(a + g)/2, prec], N[Sqrt[a g], prec]}; 
   s += p (a^2 - g^2); p += p]; N[4 a^2/(1 - s), prec]]
 
 
pi[7, 100] - N[Pi, 100]
1.2026886537*10^-86
 
pi[7, 100]
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628046852228654