Reciprocal Lucas Constant

Author

Eric W. Weisstein
March 25, 2009

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

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Sum

L[q_]:=((QPolyGamma[1,q]+Log[1-q])/(Log[q]))

L[q]//TraditionalForm

(∑_n^∞)(1/(LucasL[n]))==(∑_n^∞)(1/((((-ϕ))^(-n))+(ϕ^n)))==1.962858173

Sum[(1/((((-1))^n)(ϕ^(-n))+(ϕ^n))),{n,∞}]

Simplify[Refine[FunctionExpand[1/LucasL[n]],n∈Integers]/.(1+(Sqrt[5]))->2ϕ,n∈Integers]

NSum[(1/((((-1))^n)(ϕ^(-n))+(ϕ^n)))/.ϕ->GoldenRatio,{n,∞}]

Combined

s1=-(1/(ArcCosh[(7/2)]))(π-I (QPolyGamma[0,1-((I π)/(4 Log[GoldenRatio])),(GoldenRatio^2)]-QPolyGamma[0,1+((I π)/(4 Log[GoldenRatio])),(GoldenRatio^2)]));

s2=(1/(4 Log[GoldenRatio]))(I π-QPolyGamma[0,(1/2),(GoldenRatio^2)]+QPolyGamma[0,(1/2)-((I π)/(2 Log[GoldenRatio])),(GoldenRatio^2)]);

s1+s2//N

FullSimplify[s1+s2]

FullSimplify[((-(QPolyGamma3[0, (1/2), (ϕ^2)])+(QPolyGamma3[0, (1/2)-((I π)/(2 (csch^(-1))(2))), (ϕ^2)])+I π)/(4 log(ϕ)))-((π-I ((QPolyGamma3[0, 1-((I π)/(4 (csch^(-1))(2))), (ϕ^2)])-(QPolyGamma3[0, 1+((I π)/(4 (csch^(-1))(2))), (ϕ^2)])))/(4 log(ϕ)))]

(1/(4 ArcCsch[2]))((-1+I) π-QPolyGamma[0,(1/2),(GoldenRatio^2)]+I (QPolyGamma[0,1-((I π)/(4 ArcCsch[2])),(GoldenRatio^2)]-QPolyGamma[0,1+((I π)/(4 ArcCsch[2])),(GoldenRatio^2)])+QPolyGamma[0,(1/2)-((I π)/(2 ArcCsch[2])),(GoldenRatio^2)])/.ArcCsch[2]->Log[GoldenRatio]

(1/(4 Log[GoldenRatio]))(-QPolyGamma[0,(1/2),(GoldenRatio^2)]+QPolyGamma[0,(1/2)-((I π)/(2 Log[GoldenRatio])),(GoldenRatio^2)]+I (QPolyGamma[0,1-((I π)/(4 Log[GoldenRatio])),(GoldenRatio^2)]-QPolyGamma[0,1+((I π)/(4 Log[GoldenRatio])),(GoldenRatio^2)])+π (-1+I))//N

Numerical

NSum[(1/(LucasL[n])),{n,∞},WorkingPrecision->500,PrecisionGoal->110]//Chop

Total[Table[((1`120)/(LucasL[n])),{n,10^3}]]

N[Sum[1`2000/LucasL[n],{n,5000}],1000]

N[Sum[1`2000/LucasL[n],{n,10^4}],1000]

N[Plus@@Table[(1/(LucasL[n])),{n,10^3}],110]

V6

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

Sum[(1/(LucasL[n])),{n,∞}]

Sum[(1/(LucasL[n])),{n,k}]

V7

Sum[(1/(LucasL[n])),n]

Sum[(1/(LucasL[n])),{n,k}]//Tming

Sum[(1/(LucasL[n])),{n,∞}]//Timing

NSum[(1/(LucasL[n])),{n,∞}]

FunctionExpand[LucasL[n]]

N[1/(1/2) (1+(Sqrt[5]))]

Sum[(1/(((((1/2) (1-(Sqrt[5]))))^n)+((((1/2) (1+(Sqrt[5]))))^n))),n]

Sum[(1/(((((1/2) (1-(Sqrt[5]))))^n)+((((1/2) (1+(Sqrt[5]))))^n))),{n,k}]

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

NSum[(1/((((-1))^n)(GoldenRatio^(-n))+(GoldenRatio^n))),{n,∞}]

NSum[((Fibonacci[n])/(Fibonacci[2n])),{n,∞}]

Sum[((Fibonacci[n])/(Fibonacci[2n])),{n,∞}]//Timing

Quit

(∑_n^∞)(1/(LucasL[2 n]))==(∑_n^∞)(1/((ϕ^(-2 n))+(ϕ^(2 n))))==(∑_n^∞)((ϕ^(2 n))/((ϕ^(4 n))+1))==(1/4)(((EllipticTheta[3, 0, (1/(ϕ^2))])^2)-1)==((-π+i ((QPolyGamma3[0, 1+((i π)/(4 log(ϕ))), (ϕ^2)])-(QPolyGamma3[0, 1-((i π)/(4 log(ϕ))), (ϕ^2)])))/(4 log(ϕ)))==0.56617767581

(1/4)(((EllipticTheta[3,0,(GoldenRatio^(-2))])^2)-1)//N

Simplify[Refine[FunctionExpand[1/LucasL[2n]],n∈Integers]/.(1+(Sqrt[5]))->2ϕ,n∈Integers]

(∑_n^∞)(1/((ϕ^(-2 n))+(ϕ^(2 n))))

NSum[(1/((ϕ^(-2 n))+(ϕ^(2 n))))/.ϕ->GoldenRatio,{n,∞}]

NSum[((ϕ^(2 n))/(1+(ϕ^(4 n))))/.ϕ->GoldenRatio,{n,∞}]

Factor[(1/((ϕ^(-2 n))+(ϕ^(2 n))))]

eqn=Table[((ϕ^(2 n))/(1+(ϕ^(4 n))))==((a/((ϕ^(k n))-1))+(b/((ϕ^(l n))-1)))/.ϕ->GoldenRatio,{n,4}];

FindRoot[Evaluate[eqn//N],{{a,2},{b,1},{k,-3},{l,-2}}]

NSum[1/LucasL[2n],{n,∞}]

V7

Sum[(1/(LucasL[2n])),{n,∞}]//Timing

NSum[(1/(LucasL[2n])),{n,0,∞},WorkingPrecision->20]

NSum[(1/(LucasL[2n])),{n,∞},WorkingPrecision->20]

Sum[(1/(LucasL[2n])),n]

s=Sum[(1/(LucasL[2n])),{n,k}]//FullSimplify

Limit[s,k->0]

Limit[s,k->1]

Limit[s,k->∞]//FullSimplify

((-(1/(ArcCosh[(7/2)]))(π-I (QPolyGamma[0,1-((I π)/(2 ArcCosh[(3/2)])),(1/2) (3+(Sqrt[5]))]-QPolyGamma[0,1+((I π)/(2 ArcCosh[(3/2)])),(1/2) (3+(Sqrt[5]))]))/.ArcCosh[(3/2)]->Log[(1/2) (3+(Sqrt[5]))]/.{{(3+Sqrt[5])->2GoldenRatio^2,(2/(3+(Sqrt[5])))->1/GoldenRatio^2}}//FullSimplify)/.ArcCsch[2]->Log[GoldenRatio]//FullSimplify)/.ArcCsch[2]->Log[GoldenRatio]

N[%]

TrigToExp[ArcCosh[7/2]]//FullSimplify

Table[((1+(Sqrt[5]))/2)^n,{n,4,4}]//Expand//Factor

-(1/(4Log[GoldenRatio]))(π-I (QPolyGamma[0,1-((I π)/(4 Log[GoldenRatio])),(GoldenRatio^2)]-QPolyGamma[0,1+((I π)/(4 Log[GoldenRatio])),(GoldenRatio^2)]))//N

(∑_(n=0)^∞)(1/(LucasL[2 n+1]))==(∑_(n=0)^∞)(1/((ϕ^(2 n+1))-(ϕ^(-(2 n+1)))))==L((1/(ϕ^4)))-2 L((1/(ϕ^2)))+L((1/ϕ))==((-4 (QPolyGamma3[0, 1, (1/ϕ)])+4 (QPolyGamma3[0, 1, (1/(ϕ^2))])-(QPolyGamma3[0, 1, (1/(ϕ^4))])-log((ϕ^2)+1)+7 log(ϕ))/(4 log(ϕ)))==((-(QPolyGamma3[0, (1/2), (ϕ^2)])+(QPolyGamma3[0, (1/2)-((i π)/(2 log(ϕ))), (ϕ^2)])+i π)/(4 log(ϕ)))==1.3966804973982612`

NSum[(1/(LucasL[2n+1])),{n,0,∞}]

NSum[(1/((ϕ^(2 n+1))-(ϕ^(-(2 n+1)))))/.ϕ->GoldenRatio,{n,0,∞}]

L[(ϕ^(-1))]-2L[(ϕ^(-2))]+L[(ϕ^(-4))]/.ϕ->GoldenRatio//N

FullSimplify[L[(ϕ^(-1))]-2L[(ϕ^(-2))]+L[(ϕ^(-4))],ϕ>0]

FullSimplify[L[(ϕ^(-1))]-2L[(ϕ^(-2))]+L[(ϕ^(-4))]/.ϕ->GoldenRatio]

(1/(4 Log[ϕ]))(-Log[((-1+(ϕ^4))/(((1+ϕ))^4))]-QPolyGamma[0,1,(1/(GoldenRatio^4))]+4 QPolyGamma[0,1,(1/(GoldenRatio^2))]-4 QPolyGamma[0,1,(1/GoldenRatio)])/.ϕ->GoldenRatio//N

FullSimplify[((-1+(ϕ^4))/(((1+ϕ))^4))/.ϕ->GoldenRatio]//N

Factor[((-1+(ϕ^4))/(((1+ϕ))^4))]

FullSimplify[(((-1+ϕ) (1+(ϕ^2)))/(((1+ϕ))^3))/.ϕ->GoldenRatio]//N

FullSimplify[(((1+(ϕ^2)))/(ϕ^7))/.ϕ->GoldenRatio]//N

(1/(4 Log[ϕ]))(-Log[(((1+(ϕ^2)))/(ϕ^7))]-QPolyGamma[0,1,(1/(GoldenRatio^4))]+4 QPolyGamma[0,1,(1/(GoldenRatio^2))]-4 QPolyGamma[0,1,(1/GoldenRatio)])/.ϕ->GoldenRatio//N

(1/(4 Log[ϕ]))(7Log[ϕ]-Log[1+(ϕ^2)]-QPolyGamma[0,1,(1/(GoldenRatio^4))]+4 QPolyGamma[0,1,(1/(GoldenRatio^2))]-4 QPolyGamma[0,1,(1/GoldenRatio)])/.ϕ->GoldenRatio//N

FullSimplify[(1+(ϕ^2))/.ϕ->FunctionExpand[GoldenRatio]]

Simplify[Refine[FunctionExpand[1/LucasL[2n+1]],n∈Integers]/.(1+(Sqrt[5]))->2ϕ,n∈Integers]

NSum[((ϕ^(1+2 n))/(-1+(ϕ^(2+4 n))))/.ϕ->GoldenRatio,{n,0,∞}]

Factor[((ϕ^(1+2 n))/(-1+(ϕ^(2+4 n))))]

Apart[((ϕ^(1+2 n))/(-1+(ϕ^(2+4 n))))]

eqn=Table[(( (ϕ^(2 n+1)))/((-1+(ϕ^(2 n+1)))(1+(ϕ^(2 n+1)))))==((a/((ϕ^(k n))-1))+(b/((ϕ^(l n))-1)))/.ϕ->GoldenRatio,{n,4}];

FindRoot[Evaluate[eqn//N],{{a,3},{b,3},{k,-4},{l,-2}},MaxIterations->1000]

FindRoot[eqn,{{a,1},{b,-1},{k,1},{l,1}}]

V7

Sum[(1/(LucasL[2n+1])),{n,0,∞}]//Timing

Sum[(1/(LucasL[2n+1])),n]//Timing

Sum[(1/(LucasL[2n+1])),{n,0,k}]//FullSimplify

Limit[%,k->∞]//FullSimplify

N[%]

((1/(2 ArcCosh[(3/2)]))(I π-QPolyGamma[0,(1/2),(1/2) (3+(Sqrt[5]))]+QPolyGamma[0,(1/2)-((I π)/(ArcCosh[(3/2)])),(1/2) (3+(Sqrt[5]))])/.ArcCosh[(3/2)]->Log[(1/2) (3+(Sqrt[5]))]/.{{(3+Sqrt[5])->2GoldenRatio^2,(2/(3+(Sqrt[5])))->1/GoldenRatio^2}}//FullSimplify)/.ArcCsch[2]->Log[GoldenRatio]

N[%]