Pi Formulas

Author

Eric W. Weisstein
February 2, 2022

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

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

©2022 Wolfram Research, Inc. except for portions noted otherwise

Series

Gregory Series (π/4)==(∑_k^∞)((((-1))^(k+1))/(2 k-1))

IntegerListPlot[Table[π+(((-1))^n) (PolyGamma[0,(1/4)+(n/2)]-PolyGamma[0,(3/4)+(n/2)]),{n,50}],
AxesLabel->TraditionalForm/@{n,TraditionalForm[HoldForm[4Sum[((((-1))^(k+1))/(2k-1)),{k,n}]]]},
Epilog->{Blue,Line[{{0,#},{50,#}}]&[Pi]}]

V6

Pi/4==Plus@@HoldForm/@Table[((((-1))^(k+1))/(2k-1)),{k,5}]//TraditionalForm

4Sum[((((-1))^(k+1))/(2k-1)),{k,n}]//FullSimplify

4Sum[((((-1))^(k+1))/(2k-1)),{k,∞}]

V7

4Sum[((((-1))^(k+1))/(2k-1)),{k,n}]

4Sum[((((-1))^(k+1))/(2k-1)),{k,∞}]

4NSum[((((-1))^(k+1))/(2k-1)),{k,∞}]

(∑_(n=1)^∞)(((((-1))^(n+1))/(4 n-1))+((((-1))^(n+1))/(4 n-3)))==(π/(2 (Sqrt[2])))

Wells

V6

Sum[(-1)^(n+1)/(4n-1)+(-1)^(n+1)/(4n-3),{n,Infinity}]

FullSimplify[%]

Factor[FullSimplify[(-1)^(n+1)/(4n-1)+(-1)^(n+1)/(4n-3)]]

FunctionExpand[Sum[-((4 (((-1))^n) (-1+2 n))/((-3+4 n) (-1+4 n))),{n,∞}]]//FullSimplify

V7

Sum[(-1)^(n+1)/(4n-1)+(-1)^(n+1)/(4n-3),{n,Infinity}]

Sum[-((4 (((-1))^n) (-1+2 n))/((-3+4 n) (-1+4 n))),{n,∞}]

(∑_(n=1)^∞)((((-1))^n)/(2 n (2 n+1) (2 n+2)))==((3-π)/4)

Wells

V6

Sum[(-1)^n/(2n(2n+1)(2n+2)),{n,Infinity}]

V7

Sum[(-1)^n/(2n(2n+1)(2n+2)),{n,Infinity}]

(∑_(n=1)^∞)(1/(n^2))==((π^2)/6)

Wells

V6

Sum[1/n^2,{n,Infinity}]

V7

Sum[1/n^2,{n,Infinity}]

(∑_(n=1)^∞)(1/(((2 n-1))^2))==((π^2)/8)

Wells

V6

Sum[1/(2n-1)^2,{n,Infinity}]

V7

Sum[1/(2n-1)^2,{n,Infinity}]

(∑_(k=0)^∞)((2 (((-1))^k) (3^((1/2)-k)))/(2 k+1))==π

Crepps

V6

Sum[((2 (((-1))^k) (3^((1/2)-k)))/(1+2 k)),{k,0,∞}]

V7

Sum[((2 (((-1))^k) (3^((1/2)-k)))/(1+2 k)),{k,0,∞}]

(∑_n^∞)(cot^(-1))((Fibonacci[2 n+1]))==(π/4)

V6

Sum[ArcCot[Fibonacci[2n+1]],{n,∞}]

FullSimplify[FunctionExpand[ArcCot[Fibonacci[2n+1]]],n∈Integers&&n>0]

Sum[ArcCot[(((((2 (1+(Sqrt[5]))))^(-2 n)) ((2^(1+4 n))+(((1+(Sqrt[5])))^(4 n)) (3+(Sqrt[5]))))/(5+(Sqrt[5])))],{n,∞}]

N[{Pi/4,Sum[ArcCot[Fibonacci[2n+1]],{n,100}]},20]

V7

TimeConstrained[Sum[ArcCot[Fibonacci[2n+1]],{n,∞}],60]//Timing

(∑_(k=0)^∞)(4 k+1) (((-1))^k) ((((((2 k-1)!!)/((2 k)!!))))^3)==(2/π)

Ramanujan, Twelve Lectures (p. 7)

V6

Sum[FunctionExpand[(4k+1)(-1)^k ((2k-1)!!/(2k)!!)^3],{k,0,∞}]//FullSimplify

N[{2/π,%},20]

V7

Sum[(4k+1)(-1)^k ((2k-1)!!/(2k)!!)^3,{k,0,∞}]

((Sqrt[8])/9801) (∑_(n=0)^∞)(((26390 n+1103) (4 n)!)/((396^(4 n)) (((n!))^4)))==(1/π)

V6

FullSimplify[((Sqrt[8])/9801) Sum[(((4n)!(1103+26390n))/((((n!))^4)(396^(4n)))),{n,0,∞}]]

N[%-1/Pi,1000]

PiSeries[k_]:=Sqrt[8]/9801 Sum[(4n)!(1103+26390n)/(n!)^4/396^(4n),{n,0,k}]

Table[Simplify[PiSeries[i]],{i,0,5}]

Table[N[PiSeries[i],30],{i,0,10}]

N[Pi,30]

V7

((Sqrt[8])/9801) Sum[(((4n)!(1103+26390n))/((((n!))^4)(396^(4n)))),{n,0,∞}]

(Sqrt[8]) (∑_(n=0)^∞)(((26390 n+1103)  (2 n-1)!! (4 n-1)!!)/((99^(4 n+2)) (32^n) (((n!))^3)))==(1/π)

V6

PiSeries2[k_]:=Sqrt[8]Sum[(4n)!(1103+26390n)(2n-1)!!(4n-1)!!/99^(4n+2)/32^n/(n!)^3,{n,0,k}]

Sqrt[8]Sum[(4n)!(1103+26390n)(2n-1)!!(4n-1)!!/99^(4n+2)/32^n/(n!)^3,{n,0,k}]

1/Table[PiSeries2[i],{i,0,6}]

1/Table[N[PiSeries2[i],30],{i,0,10}]

V7

(Sqrt[8])Sum[(1103+26390 n) (2 n-1)!! (4 n-1)!!/99^(4 n+2)/32^n/(n!)^3,{n,0,∞}]

FullSimplify[%]

N[%]

1/π//N

(((∑_(n=1)^∞)((3 (-885673181 (n^5)+3125347237 (n^4)-2942969225 (n^3)+1031962795 (n^2)-196882274 n+10996648))/((2^(n-1)) (Binomial[7 n, 2 n])))-20379280)/740025)==π

V6

P[n_]:=-885673181n^5+3125347237n^4-2942969225n^3+1031962795n^2-196882274n+10996648

(1/740025) (NSum[((3P[n])/(Binomial[7n,2n](2^(n-1)))),{n,100}]-20379280)

pin[k_]:=1/740025 (Sum[3P[n]/Binomial[7n,2n]/2^(n-1),{n,k}]-20379280)

Table[pin[i],{i,10}]//N

pi1=N[12/Sqrt[130] Log[(2+Sqrt[5])(3+Sqrt[13])/Sqrt[2]],50]

pi2=N[12/Sqrt[130] Log[(3+Sqrt[13])(Sqrt[8]+Sqrt[10])/2],50]

pi1-pi2

pi3=N[24/Sqrt[142] Log[(Sqrt[10+11Sqrt[2]]+Sqrt[10+7Sqrt[2]])/2],50]

pi4=N[12/Sqrt[310]Log[(3+Sqrt[5])(2+Sqrt[2])/4 (5+2Sqrt[10]+Sqrt[61+20Sqrt[10]])],50]

pi5=N[4/Sqrt[522]Log[((5+Sqrt[29])/Sqrt[2])^3(5Sqrt[29]+11Sqrt[6])*
  (Sqrt[(9+3Sqrt[6])/4]+Sqrt[(5+3Sqrt[6])/4])^6],50]

N[{pi2,pi3,pi4,pi5}-Pi,50]

V7

P[n_]:=-885673181n^5+3125347237n^4-2942969225n^3+1031962795n^2-196882274n+10996648

(1/740025) (NSum[((3P[n])/(Binomial[7n,2n](2^(n-1)))),{n,∞}]-20379280)

(1/740025) (Sum[((3P[n])/(Binomial[7n,2n](2^(n-1)))),{n,∞}]-20379280)

Simplify[%]

12 (∑_(k=0)^∞)(((((-1))^k) (545140134 k+13591409) (6 k)!)/((640320^(3 k+(3/2))) (3 k)! (((k!))^3)))==(1/π)

Chudnovsky

V6

12 Sum[(((((-1))^k)(6k)!(13591409+545140134k))/((3k)!(((k!))^3)(640320^(3k+3/2)))),{k,0,∞}]

FunctionExpand[%]

N[%-1/Pi,1000]

V7

12 Sum[(((((-1))^k)(6k)!(13591409+545140134k))/((3k)!(((k!))^3)(640320^(3k+3/2)))),{k,0,∞}]

(∑_(n=0)^∞)(1/16) (42 n+5) (((((Binomial[2 n, n])/(16^n))))^3)==(1/π)

V6

Sum[(((((Binomial[2n,n])/(16^n))))^3)((42n+5)/16),{n,0,∞}]//FullSimplify

N[{1/Pi,%},20]

FullSimplify[(((((Binomial[2n,n])/(16^n))))^3)((42n+5)/16),n∈Integers&&n≥0]

V7

Sum[(((((Binomial[2n,n])/(16^n))))^3)((42n+5)/16),{n,0,∞}]

(4/π)==(∑_(n=0)^∞)(((6 n+1) ((Subscript[((1/2)), n])^3))/((4^n) (((n!))^3)))

Ramanujan Series

V6

{#,N[#,20]}&[{(4/Pi),Sum[(((6n+1)Pochhammer[1/2,n]^3)/((4^n)(((n!))^3))),{n,0,∞}]}//FullSimplify]

{#,N[#,20]}&[{(16/Pi),Sum[(((42n+5)Pochhammer[1/2,n]^3)/((64^n)(((n!))^3))),{n,0,∞}]}//FullSimplify]

V7

Sum[(((6n+1)Pochhammer[1/2,n]^3)/((4^n)(((n!))^3))),{n,0,∞}]

FullSimplify[%]

N[%]

4/Pi//N

(∑_(n=0)^∞)((((-1))^n)/(2^(20 n))) (820 (n^2)+180 n+13) ((Binomial[2 n, n])^5)==(128/(π^2))

W-Z

V6

s=Sum[((((-1))^n)/(2^(20n)))Binomial[2n,n]^5(820n^2+180n+13),{n,0,∞}]

N[{s,128/Pi^2},20]

Expand[s]

TraditionalForm[%]

V7

Sum[((((-1))^n)/(2^(20n)))Binomial[2n,n]^5(820n^2+180n+13),{n,0,∞}]

FullSimplify[%]

N[%]

128/π^2//N

(∑_(k=0)^∞)(1/(16^k))(((4 (((z+1))^(8 k+1)))/(8 k+1))-((2 (((z+1))^(8 k+4)))/(8 k+4))-((((z+1))^(8 k+5))/(8 k+5))-((((z+1))^(8 k+6))/(8 k+6)))==4 (tan^(-1))(z)+2log(((-(z^2)-2 z+1)/((z^2)+1)))+π

Amadchik-Wagon BBP

V6

Sum[(1/(16^k))(((4(((z+1))^(8k+1)))/(8k+1))-((2(((z+1))^(8k+4)))/(8k+4))-((((z+1))^(8k+5))/(8k+5))-((((z+1))^(8k+6))/(8k+6))),{k,0,∞}]//FullSimplify

Table[%-(π+4ArcTan[z]+2Log[((-(z^2)-2z+1)/((z^2)+1))])/.z->RandomComplex[],{20}]//Chop//Union

FullSimplify[Re[(-2 ArcCoth[(4/(((1+z))^4))]+4 (1+z) Hypergeometric2F1[(1/8),1,(9/8),(1/16) (((1+z))^8)]-(1/5) (((1+z))^5) Hypergeometric2F1[(5/8),1,(13/8),(1/16) (((1+z))^8)]-(1/6) (((1+z))^6) Hypergeometric2F1[(3/4),1,(7/4),(1/16) (((1+z))^8)])-(π+4ArcTan[z]+2Log[((1-2z-(z^2))/((z^2)+1))])/.z->x+I y],(x|y)∈Reals]

ContourPlot[-π-2 Re[ArcCoth[(4/(((1+x+I y))^4))]]-4 Re[ArcTan[x+I y]]+4 Re[(1+x+I y) Hypergeometric2F1[(1/8),1,(9/8),(1/16) (((1+x+I y))^8)]]-(1/5) Re[(((1+x+I y))^5) Hypergeometric2F1[(5/8),1,(13/8),(1/16) (((1+x+I y))^8)]]-(1/6) Re[(((1+x+I y))^6) Hypergeometric2F1[(3/4),1,(7/4),(1/16) (((1+x+I y))^8)]]-2 Re[Log[((1-2 (x+I y)-(((x+I y))^2))/(1+(((x+I y))^2)))]],{x,-5,5},{y,-5,5},ContourLines->False,ColorFunction->Hue,PlotPoints->200,FrameLabel->TraditionalForm/@{x,y}]//Timing

Plot[Evaluate[Table[-π-2 Re[ArcCoth[(4/(((1+x+I y))^4))]]-4 Re[ArcTan[x+I y]]+4 Re[(1+x+I y) Hypergeometric2F1[(1/8),1,(9/8),(1/16) (((1+x+I y))^8)]]-(1/5) Re[(((1+x+I y))^5) Hypergeometric2F1[(5/8),1,(13/8),(1/16) (((1+x+I y))^8)]]-(1/6) Re[(((1+x+I y))^6) Hypergeometric2F1[(3/4),1,(7/4),(1/16) (((1+x+I y))^8)]]-2 Re[Log[((1-2 (x+I y)-(((x+I y))^2))/(1+(((x+I y))^2)))]],{y,0,5,1}]],{x,-4,4},PlotRange->All,PlotStyle->{Red,Orange,Yellow,Green,Blue}]

V7

Sum[(1/(16^k))(((4(((z+1))^(8k+1)))/(8k+1))-((2(((z+1))^(8k+4)))/(8k+4))-((((z+1))^(8k+5))/(8k+5))-((((z+1))^(8k+6))/(8k+6))),{k,0,∞}]

FullSimplify[%]

FullSimplify[FunctionExpand[%]]

(∑_(k=0)^∞)(((((-1))^k) (4 k+1) ((((2 k-1)!!))^3))/((((2 k)!!))^3))==(∑_(k=0)^∞)(((((-1))^k) (4 k+1) ((Γ(k+(1/2)))^3))/((π^(3/2)) ((Γ(k+1))^3)))==(2/π)

Sum[(((((-1))^k)(4k+1)(((2k-1)!!)^3))/(((2k)!!)^3)),{k,0,∞}]

Sum[(((((-1))^k)(4k+1)((Gamma[k+(1/2)])^3))/((π^(3/2))((Gamma[k+1])^3))),{k,0,∞}]

72 (∑_(n=1)^∞)(1/(n ((e^(π n))-1)))-96 (∑_(n=1)^∞)(1/(n ((e^(2 π n))-1)))+24 (∑_(n=1)^∞)(1/(n ((e^(4 π n))-1)))==π

Plouffe

V6

20 Apr 2006

72Sum[(1/(n(Exp[π n]-1))),{n,∞}]-96Sum[(1/(n(Exp[2π n]-1))),{n,∞}]+24Sum[(1/(n(Exp[4π n]-1))),{n,∞}]

N[%,20]

V7

72Sum[(1/(n(Exp[π n]-1))),{n,∞}]-96Sum[(1/(n(Exp[2π n]-1))),{n,∞}]+24Sum[(1/(n(Exp[4π n]-1))),{n,∞}]

72NSum[(1/(n(Exp[π n]-1))),{n,∞}]-96NSum[(1/(n(Exp[2π n]-1))),{n,∞}]+24NSum[(1/(n(Exp[4π n]-1))),{n,∞}]

4-4 (∑_(k=0)^∞)(((((-1))^k) (2^(4 k)) (((2 k+1))^3) (40 (k^2)+96 k+57) (({{2 k}, {k}})))/((((64 (k^3)+144 (k^2)+92 k+15))^2) (((({{4 k}, {2 k}})))^2)))==π

Lupas

V6

Sum[(((-1))^k)(((((Pochhammer[x,k])/(Pochhammer[x+2y,k]))))^2),{k,0,∞}]/.{x->1/2,y->1}

N[%9]

Apart//@Solve[y==-(9/8) (-4+p),p]

expr=(y/2)(((-1)^k)/(16^k))(((Pochhammer[2y+1,2k])^3)/(((Pochhammer[y+1,k])^2)((Pochhammer[x+2y,2k+1])^2)))((10(((k+y))^2)+(6x+1)(k+y)+(x^2))/(k+y))/.{x->1/2,y->1}

NSum[expr,{k,0,∞}]

FunctionExpand[expr]

NSum[expr,{k,0,∞}]

((9 (((-1))^k) (57+96 k+40 (k^2)) ((Gamma[(3/2)+k])^2) Gamma[2+2 k])/(32 ((Gamma[(7/2)+2 k])^2)))/.{Gamma[(3/2)+k]->(((2(k+1)-1)!!(Sqrt[π]))/(2^(k+1))),Gamma[2+2k]->(2+2k)(1+2k)Gamma[2k],Gamma[(7/2)+2k]->(((2(3+2k)-1)!!(Sqrt[π]))/(2^(2k+3)))}//FullSimplify

Solve[(3/2)+k==1/2+n,n]

Solve[(7/2)+2k==1/2+n,n]

Simplify[FunctionExpand[(((Binomial[2k,k])/((Binomial[4k,2k])^2))/(( ((Gamma[(3/2)+k])^2) Gamma[2+2 k])/( ((Gamma[(7/2)+2 k])^2))))]]

(((2^(-4 (1+k))) (((1+4 k))^2) (((3+4 k))^2) (((5+4 k))^2))/(((1+2 k))^3))

Simplify[((9 (((-1))^k) (57+96 k+40 (k^2)) ((Gamma[(3/2)+k])^2) Gamma[2+2 k])/(32 ((Gamma[(7/2)+2 k])^2)))(((Binomial[2k,k])/((Binomial[4k,2k])^2))/(( ((Gamma[(3/2)+k])^2) Gamma[2+2 k])/( ((Gamma[(7/2)+2 k])^2))))(1/(((2^(-4 (1+k))) (((1+4 k))^2) (((3+4 k))^2) (((5+4 k))^2))/(((1+2 k))^3)))]

NSum[%,{k,0,∞}]

Factor[((9 (((-1))^k) (2^(-1+4 k)) (((1+2 k))^3) (57+96 k+40 (k^2)) Binomial[2 k,k])/((((15+92 k+144 (k^2)+64 (k^3)))^2) ((Binomial[4 k,2 k])^2)))]

4-4Sum[(((((-1))^k) (2^(4 k)) (((1+2 k))^3) (57+96 k+40 (k^2)) Binomial[2 k,k])/((((15+92 k+144 (k^2)+64 (k^3)))^2) ((Binomial[4 k,2 k])^2))),{k,0,∞}]

N[%]

Sum[((((-1))^(j+1))/(2j-1)),{j,n}]//FullSimplify

(1/4) (π+(((-1))^n) (PolyGamma[0,(1/4)+(n/2)]-PolyGamma[0,(3/4)+(n/2)]))//FunctionExpand

V7

4-4Sum[(((((-1))^k) (2^(4 k)) (((1+2 k))^3) (57+96 k+40 (k^2)) Binomial[2 k,k])/((((15+92 k+144 (k^2)+64 (k^3)))^2) ((Binomial[4 k,2 k])^2))),{k,0,∞}]

TimeConstrained[FullSimplify[%],60]//Timing

N[%%]

(π/4)==(∑_(j=1)^n)((((-1))^(j+1))/(2 j-1))+(1/16) (((-1))^n) (2 n-1)! (∑_(k=0)^∞)(1/(16^k))[-(4/(Subscript[(8 k+3), 2 n]))-(4/(Subscript[(8 k+4), 2 n]))-(2/(Subscript[(8 k+5), 2 n]))+(1/(Subscript[(8 k+7), 2 n]))+(1/(Subscript[(8 k+8), 2 n]))+(8/(Subscript[(8 k+1), 2 n]))]

V6

TraditionalForm[sum=Sum[((((-1))^(j+1))/(2j-1)),{j,n}]+(((-1))^n)(((2n-1)!)/16)Sum[(1/(16^k))((8/(Pochhammer[8k+1,2n]))-(4/(Pochhammer[8k+3,2n]))-(4/(Pochhammer[8k+4,2n]))-(2/(Pochhammer[8k+5,2n]))+(1/(Pochhammer[8k+7,2n]))+(1/(Pochhammer[8k+8,2n]))),{k,0,∞}]]

FullSimplify[Table[sum,{n,4}]]//Timing

V7

Sum[((((-1))^(j+1))/(2j-1)),{j,n}]+(((-1))^n)(((2n-1)!)/16)Sum[(1/(16^k))((8/(Pochhammer[8k+1,2n]))-(4/(Pochhammer[8k+3,2n]))-(4/(Pochhammer[8k+4,2n]))-(2/(Pochhammer[8k+5,2n]))+(1/(Pochhammer[8k+7,2n]))+(1/(Pochhammer[8k+8,2n]))),{k,0,∞}]

FullSimplify[%,n∈Integers&&n>0]

Table[%222,{n,1.,10}]//Chop

π/4.

12 (∑_(n=0)^∞)((((-1))^n) (107578229802750 n+13773980892672 (Sqrt[61])+212175710912 (Sqrt[61])+1657145277365) (6 n)!)/((((5280 (30303 (Sqrt[61])+236674)))^(3^(n+(1/2)))) (((n!))^3) (3 n)!)==(1/π)

V7

AA=212175710912Sqrt[61]+1657145277365;
BB=13773980892672Sqrt[61]+107578229802750;
CC=(5280(236674+30303Sqrt[61]))^3;

1/(12 Sum[(-1)^n(6n)!(AA+BB n)/(n!)^3/(3n)!/CC^(n+1/2),{n,0,Infinity}])

FullSimplify[%]

N[%]

(6/7) (Sqrt[3]) ((1/3) (∑_(n=0)^∞)((843 n+4607)/((n+5) (3 n+7) (3 n+22)))-(655999/248976)-(1/2) 7 log(3))==π

Alexander Povolotsky
Sequence Fanatics Discussion list
Sun, 7 Dec 2008 08:39:45 -0500

(6/7)((1/3)Sum[((843n+4607)/((n+5)(3n+7)(3n+22))),{n,0,∞}]- 655999/248976-7/2*Log[3])Sqrt[3]

FullSimplify[%]

Products

(2/((∏_(n=2)^∞)(((((-1))^( ((Subscript[p, n])-1)/2))/(Subscript[p, n]))+1)))

Blatner's Product

Plot

With[{max=5000},
ListPlot[2/FoldList[Times,1.,Table[1+Sin[Pi Prime[n]/2]/Prime[n],{n,max}]],
Joined->True,PlotRange->{Automatic,{3.13,3.15}},
AxesOrigin->{0,Pi},AxesLabel->{n,(Subscript[π, n])},
PlotStyle->Red,
Epilog->{Blue,Thickness[.004],Line[{{0,Pi},{max,Pi}}],Text[Pi,{200,Pi+.001}]}]
]

V6

ps=Prime/@Range[10^7];//Timing

p1=N[Select[ps,Mod[#,4]==1&],20];
p3=N[Select[ps,Mod[#,4]==3&],20];

Plus@@Length/@{p1,p3}

2/(Times@@((1+1/#)&/@p1)Times@@((1-1/#)&/@p3))

2/Times@@Table[1.+(-1)^((Prime[n]-1)/2)/Prime[n],{n,2,#}]&/@{10^2,10^3,10^4,10^5}

V7

2/Product[1+(-1)^((Prime[n]-1)/2)/Prime[n],{n,2,∞}]

2/Product[1.+(-1)^((Prime[n]-1)/2)/Prime[n],{n,2,10^6}]

((2 sin((2/4) π (n-3)))/((n-3) (∏_(k=0)^∞)cos(((π (n-3))/(4 (2^k))))))==π

Beckmann

V6

pi[n_]:=Module[{b=(n-3)Pi/4},
2/(n-3) Sin[2b]/Product[Cos[b/2^k],{k,0,Infinity}]]

pi[n]

Table[pi[n],{n,8}]//N

V7

With[{b=(n-3)Pi/4},2/(n-3) Sin[2b]/Product[Cos[b/2^k],{k,0,Infinity}]]

FullSimplify[%]

(((∏_(n=1)^∞)((1/(4 (n^2)-1))+1))/((∑_(n=1)^∞)(1/(4 (n^2)-1))))

Sondow

V6

Product[1+1/(4n^2-1),{n,1,Infinity}]

Sum[1/(4n^2-1),{n,1,Infinity}]

((Product[1+1/(4n^2-1),{n,1,Infinity}])/(Sum[1/(4n^2-1),{n,1,Infinity}]))

V7

Product[1+1/(4n^2-1),{n,1,Infinity}]

Sum[1/(4n^2-1),{n,1,Infinity}]

((Product[1+1/(4n^2-1),{n,1,Infinity}])/(Sum[1/(4n^2-1),{n,1,Infinity}]))

2 (∏_n^∞)((((2 n))^2)/((2 n-1) (2 n+1)))

Wallis

V6

2 Product[ (2n)^2/(2n-1)/(2n+1),{n,∞}]

Table[(2^(2n)/Binomial[2n,n]/Sqrt[n])^2//N,{n,100,2000,100}]

<<NumericalMath`NLimit`

NLimit[16^n/Binomial[2n,n]^2/n,n->Infinity]

Simplify[(2n!/n!/n!)^2/n]

FullSimplify[2^(4n)/Binomial[2n,n]^2/n]

NLimit[n Gamma[n]^2/Gamma[n+1/2]^2,n->Infinity]

V7

2 Product[ (2n)^2/(2n-1)/(2n+1),{n,∞}]

Limit[16^n/Binomial[2n,n]^2/n,n->Infinity]

Limit[n Gamma[n]^2/Gamma[n+1/2]^2,n->Infinity]

Integrals

Backhouse

Integrate[(((x^m)(((1-x))^n))/(1+(x^2))),{x,0,1},Assumptions->m>-1&&n>-1]

FullSimplify[%]

<<Utilities`Simplify`

Table[(2^(-1-m-n)) (Sqrt[π]) Gamma[1+m] Gamma[1+n] HypergeometricPFQRegularized[{1,((1+m)/2),((2+m)/2)},{(1/2) (2+m+n),(1/2) (3+m+n)},-1],{m,10},{n,4}]//LogExpand//Factor//TableForm//TraditionalForm

24 (∫_0^(1/4))(Sqrt[x-(x^2)])dx+((3 (Sqrt[3]))/4)==π

Newton

V6

Simplify[3Sqrt[3]/4+24Integrate[Sqrt[x-x^2],{x,0,1/4}]]

N[%]

i=Integrate[Sqrt[x-x^2],x]

s=Series[i,{x,0,10}]-(i/.x->0)

s=Series[i,{x,0,50}]-(i/.x->0);

V7

Simplify[3Sqrt[3]/4+24Integrate[Sqrt[x-x^2],{x,0,1/4}]]

Newton's method, B&B 2003, p. 105

Solve[A==pi/24-Sqrt[3]/32,pi]

Integrate[Sqrt[x-x^2],{x,0,1/4}]

Simplify[(3/4) ((Sqrt[3])+32 %)]

<<DiscreteMath`RSolve`

FullSimplify[SeriesTerm[Sqrt[1-x],{x,0,n}],n∈Integers&&n>0]

Integrate[-Sqrt[x]((Gamma[-(1/2)+n])/(2 (Sqrt[π]) n!))(x^n),{x,0,1/4},Assumptions->n>=0]

(3/4) ((Sqrt[3])-32 (Sum[(((2^(-3-2 n)) Gamma[-(1/2)+n])/((3+2 n) (Sqrt[π]) n!)),{n,0,∞}]))//FullSimplify

π==(22/7)-(∫_0^1)(((x^4) (((1-x))^4))/((x^2)+1))dx

(22/7)-Integrate[(((x^4)(((1-x))^4))/(1+(x^2))),{x,0,1}]

π==(355/113)-(1/3164)(∫_0^1)(((x^8) (((1-x))^8) (816 (x^2)+25))/((x^2)+1))dx

Borwein et al. (Experimental Math in Action, Ch. 1)

(355/113)-(1/3164)Integrate[(((x^8)(((1-x))^8)(25+816(x^2)))/(1+(x^2))),{x,0,1}]//Simplify

Nested Radicals

Munkhammer

2 Sum[Sqrt[(Sqrt[1-((n-1)/m)^2]-Sqrt[1-(n/m)^2])^2+1/m^2],{n,m}]

PowerExpand[Together//@(Sqrt[((((Sqrt[1-(((((n-1)/m)))^2)])-(Sqrt[1-((((n/m)))^2)])))^2)+(1/(m^2))])]

Limit[%,m->Infinity]

N[2 Sum[Sqrt[(Sqrt[1-((n-1)/m)^2]-Sqrt[1-(n/m)^2])^2+1/m^2],{n,m}]/.m->100,20]

N[2 Sum[Sqrt[(Sqrt[1-((n-1)/m)^2]-Sqrt[1-(n/m)^2])^2+1/m^2],{n,m}]/.m->1000,20]

N[2+8Sum[Sqrt[1-k^2/2/n^2]-1/Sqrt[2],{k,n}]/n/Sqrt[2]/. n->1000,20]

N[2+8Sum[Sqrt[1-k^2/2/n^2]-1/Sqrt[2],{k,n}]/n/Sqrt[2]/. n->10000,20]//Timing

pi[n_]:=2^(n+1)Nest[Sqrt[(#/2)^2+(1-Sqrt[1-(#/2)^2])^2]&,Sqrt[2],n]

Simplify[Table[pi[n],{n,5}]]

N[%]

f[0]:=Sqrt[2]
f[n_]:=Sqrt[(f[n-1]/2)^2+(1-Sqrt[1-(f[n-1]/2)^2])^2]

Table[2^(n+1) f[n]//N,{n,6}]

N[Pi,20]

N[pi[12],20]

Vieta

Rest[NestList[Sqrt[1/2+#/2]&,0,6]]

Times@@%

N[Times@@%-2/Pi,20]

Remove[a,b];

a[1]:=Sqrt[1/2]
a[n_]:=a[n]=Sqrt[1/2+1/2a[n-1]]
b[1]:=a[1]
b[n_]:=b[n]=a[n]b[n-1]

b[3]

a[3]

b[3]

Table[b[n],{n,12}]//N

2./Pi

Limits

(Limit2Arg[n cos(((π (n-2))/(2 n))), n, ∞])==π

Limit[n Cos[π(n-2)/2/n],n->∞]

(Limit2Arg[(((((2 (((-1))^(n+1)) (2n)!)/((2^(2n)) (BernoulliB[2n])))))^(1/(2n))), n, ∞])==π

https://arxiv.org/abs/2201.12601

DiscreteAsymptotic[BernoulliB[2 n],n->∞]

Limit[(((((2 (((-1))^(n+1))(2 n)!)/(% (2^(2 n))))))^(1/(2 n))),n->Infinity]

Limit[(((((2 (((-1))^(n+1))(2 n)!)/((2^(2 n))BernoulliB[2 n]))))^(1/(2 n))),n->Infinity]//Timing

Block[{$MaxExtraPrecision =1000},With[{n=1000},N[Pi-(((((2n!(((-1))^(n/2+1)))/((2^n)BernoulliB[n]))))^(1/n)),1000]]]//N

pts=Table[{2n,(((((2(2n)!(((-1))^(n+1)))/((2^(2n))BernoulliB[2n]))))^(1/(2n)))},{n,10}]//FullSimplify

ListPlot[pts,Filling->Axis,Epilog->{Dashing[.02],Red,Line[{{0,Pi},{20,Pi}}]},PlotRange->{Automatic,{2,Automatic}}]

(Limit2Arg[(((((2 (((-1))^(n+1)) (2n)!)/((2^(2n)) (BernoulliB[2n])(1-(2^(-n)))(1-(3^(-n)))(1-(5^(-n)))(1-(7^(-n)))))))^(1/(2n))), n, ∞])==π

https://arxiv.org/abs/2201.12601

DiscreteAsymptotic[BernoulliB[2 n],n->∞]

Limit[(((((2 (((-1))^(n+1))(2 n)!)/((2^(2 n))%(1-(2^(-n)))(1-(3^(-n)))(1-(5^(-n)))(1-(7^(-n)))))))^(1/(2 n))),n->Infinity]

Limit[(((((2 (((-1))^(n+1))(2 n)!)/((2^(2 n))BernoulliB[2 n](1-(2^(-n)))(1-(3^(-n)))(1-(5^(-n)))(1-(7^(-n)))))))^(1/(2 n))),n->Infinity]//Timing

(Limit2Arg[((((((2^(2 n+2)) (((-1))^n) (2 n)!)/(EulerE[2 n]))))^(1/(2 n+1))), n, ∞])

https://arxiv.org/abs/2201.12601

DiscreteAsymptotic[EulerE[2 n],n->∞]

Limit[((((((2^(2n+2)) (((-1))^n)(2 n)!)/(%))))^(1/(2 n+1))),n->Infinity]

Block[{$MaxExtraPrecision =1000},With[{n=100},N[Pi-((((((2^(2n+2)) (((-1))^n)(2 n)!)/(EulerE[2n]))))^(1/(2n+1))),1000]]]//N

pts=Table[{2n,((((((2^(2n+2)) (((-1))^n)(2 n)!)/(EulerE[2n]))))^(1/(2n+1)))},{n,10}]//FullSimplify

ListPlot[pts,Filling->Axis,Epilog->{Dashing[.02],Red,Line[{{0,Pi},{20,Pi}}]},PlotRange->All]

(Limit2Arg[((((((2^(2 n+2)) (((-1))^n) (2 n)!)/(EulerE[2 n]))(1-(1/(3^(2n+1))))))^(1/(2 n+1))), n, ∞])

https://arxiv.org/abs/2201.12601

DiscreteAsymptotic[EulerE[2 n],n->∞]

Limit[((((((2^(2n+2)) (((-1))^n)(2 n)!)/(%))(1-(1/(3^(2 n+1))))))^(1/(2 n+1))),n->Infinity]

(Limit2Arg[-(((n+1) (2 n+1) (BernoulliB[2 n]))/(2 (BernoulliB[2 n+2]))), n, ∞])==(π^2)

https://arxiv.org/abs/2201.12601

DiscreteAsymptotic[((BernoulliB[2 n])/(BernoulliB[2 n+2])),n->∞]

Table[-((BernoulliB[2 n](n+1)(2n+1))/(2BernoulliB[2 n+2])),{n,10}]

N[%]

Pi^2//N

(Limit2Arg[-((8 (n+1) (2 n+1) (EulerE[2 n]))/(EulerE[2 n+2])), n, ∞])==(π^2)

https://arxiv.org/abs/2201.12601

DiscreteAsymptotic[((EulerE[2 n])/(EulerE[2 n+2])),n->∞]

Table[-((8EulerE[2 n](n+1)(2n+1))/(EulerE[2 n+2])),{n,10}]

N[%]

Inequalities

(((1+(1/π)))^(π+1))<π

Digit identities

Terr

{3,1,4}==Mod[{1,5,9}+{2,6,5},10]

Digit Extraction

RSolve[{x[n]==16x[n-1]+((120(n^2)-89n+16)/(512(n^4)-1024(n^3)+712(n^2)-206n+21)),x[1]==1},x[n],n]

x[0]:=0
x[n_]:=FractionalPart[16x[n-1]+((120(n^2)-89n+16)/(512(n^4)-1024(n^3)+712(n^2)-206n+21))]

Table[Floor[16x[n]],{n,11}]

RealDigits[π-3,16,10]