Pi Iterations

Author

Eric W. Weisstein
April 16, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/PiIterations.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PiIterations.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Brent-Salamin

a[0]:=1
b[0]:=1/Sqrt[2]
a[n_]:=a[n]=1/2(a[n-1]+b[n-1])
b[n_]:=b[n]=Sqrt[a[n-1]b[n-1]]
c[n_]:=1/2(a[n-1]-b[n-1])
d[n_]:=a[n]^2-b[n]^2

N[Table[4ArithmeticGeometricMean[1,1/Sqrt[2]]^2/(1-Sum[2^(j+1) c[j]^2,{j,n}]),{n,5}],30]

N[Table[4ArithmeticGeometricMean[1,1/Sqrt[2]]^2/(1-Sum[2^(j+1) d[j],{j,n}]),{n,5}],30]

Borwein Quadratic

x[0]:=(Sqrt[2])
pi[0]:=2+Sqrt[2]
y[1]:=(2^(1/4))
x[n_]:=(1/2)((Sqrt[x[n-1]])+(1/(Sqrt[x[n-1]])))
y[n_]:=((y[n-1](Sqrt[x[n-1]])+(1/(Sqrt[x[n-1]])))/(y[n-1]+1))
pi[n_]:=pi[n-1]((x[n]+1)/(y[n]+1))

N[pi/@Range[4]-Pi,50]//N

RootReduce[pi/@Range[2]]

Salamin π/ln 2

Something's wrong here...

FullSimplify[k (2^(-k/4))(Sum[(2^(-k Binomial[n,2])),{n,∞}])^2,k>0]

f[k_]:=(1/4) k ((EllipticTheta[2,0,(2^(-k/2))])^2)

FullSimplify[((f[n])/(f[2n]))]

(Subscript[g, 0])=((f[n])/(f[2n]))

(Subscript[g, k])=(Sqrt[(1/2)((Subscript[g, k-1])+(1/(Subscript[g, k-1])))])

π = 2Log[2]f[n]Product[g[k],{k,∞}]

Cubically Convergent to Nearest Multiple of Pi

FixedPointList[#+Sin[#]&,N[23,20]]

%/Pi

g[x_]:=FixedPoint[#+Sin[#]&,x]

Plot[g[x]/Pi,{x,-10,10},PlotStyle->Red]

Quartically Convergent

y[0]:=(Sqrt[2])-1
α[0]:=6-4(Sqrt[2])
y[n_]:=((1-(((1-((y[n-1])^4)))^(1/4)))/(1+(((1-((y[n-1])^4)))^(1/4))))
α[n_]:=(((1+y[n]))^4)α[n-1]-(4^n)2 y[n](1+y[n]+((y[n])^2))

N[α/@Range[0,3]-(1/π),200]//N

Special case r==2

(SuperStar[λ])[4]=((Sqrt[2])-1)^2
α[4]=2((Sqrt[2])-1)^2

Expand[2((Sqrt[2])-1)^2]

Borwein Quntically

x=(5/(s[n-1]))-1;

y=(((x-1))^2)+7

z=(((x/2(y+Sqrt[y^2-4x^3])))^(1/5))

(25/((((z+(x/z)+1))^2)s[n-1]))

s[0]:=5((Sqrt[5])-2)
α[0]:=1/2
s[n_]:=25/((((1+((2^(1/5)) (-1+(5/(s[-1+n]))))/((((7+√((((7+(((-2+(5/(s[-1+n]))))^2)))^2)-4 (((-1+(5/(s[-1+n]))))^3))+(((-2+(5/(s[-1+n]))))^2)) (-1+(5/(s[-1+n])))))^(1/5))+(1/(2^(1/5)))(((((7+√((((7+(((-2+(5/(s[-1+n]))))^2)))^2)-4 (((-1+(5/(s[-1+n]))))^3))+(((-2+(5/(s[-1+n]))))^2)) (-1+(5/(s[-1+n])))))^(1/5)))))^2) s[-1+n])

α[n_]:=((s[n-1])^2)α[n-1]-(5^(n-1))((((s[n-1])^2)-5)/2+Sqrt[s[n-1](s[n-1]^2-2s[n-1]+5)])

N[α/@Range[3]-(1/π),200]//N

Woon

a[0]:=1
a[n_]:=Csc[(π/(2^(n+1)))]

MapIndexed[((2^(1+#2[[1]]))/(#1))&,a/@Range[5]]

N[%-Pi,200]//N

b[0]:=1
b[n_]:=Sqrt[1+(Plus@@(b/@Range[0,n-1]))^2]

N[Table[a[n]-b[n],{n,10}],20]//Chop