Reciprocal Fibonacci Constant

Author

Eric W. Weisstein
January 2, 2009

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

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

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

Sum

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

L[q]//TraditionalForm

(Subscript[P, F])==(∑_n^∞)(1/(Fibonacci[n]))==(∑_n^∞)((Sqrt[5])/((ϕ^(-n))-(((-ϕ))^n)))==3.35988566

(((Sqrt[5]) (2 (QPolyGamma3[0, 1, (1/(ϕ^4))])-4 (QPolyGamma3[0, 1, (1/(ϕ^2))])+log(5)))/(8 log(ϕ)))+(((Sqrt[5]) ((EllipticTheta[2, 0, (1/(ϕ^2))])^2))/4)

s1=((Sqrt[5])/(4 Log[GoldenRatio]))(I π-QPolyGamma[0,1,(GoldenRatio^2)]+QPolyGamma[0,1-((I π)/(2 Log[GoldenRatio])),(GoldenRatio^2)]);
s2=(1/4) (Sqrt[5]) ((EllipticTheta[2,0,(1/2) (3-(Sqrt[5]))])^2);

s1+s2

N[s1+s2]

Combined

PF=(1/4) (Sqrt[5]) (((-4 QPolyGamma[0,1,(GoldenRatio^2)]+2 QPolyGamma[0,1,(GoldenRatio^4)]+Log[5])/(2 Log[GoldenRatio]))+((EllipticTheta[2,0,(1/(GoldenRatio^2))])^2));

PF//TraditionalForm

N[PF,30]

Numerical

NSum[1/Fibonacci[n],{n,∞},WorkingPrecision->50]

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

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

q-Sum

With[{ϕ=1/GoldenRatio},NSum[((Sqrt[5])/((ϕ^(-n))-(((-ϕ))^n))),{n,∞}]]

With[{ϕ=1/GoldenRatio},(Sqrt[5])NSum[((ϕ^n)/(1-(((-1))^n)(ϕ^(2n)))),{n,∞}]]

{Refine[((ϕ^n)/(1-(((-1))^n)(ϕ^(2n)))),Mod[n,2]==1],Refine[((ϕ^n)/(1-(((-1))^n)(ϕ^(2n)))),Mod[n,2]==0]}

With[{ϕ=1/GoldenRatio},
(Sqrt[5])(NSum[((ϕ^n)/(1+(ϕ^(2 n)))),{n,1,∞,2}]+NSum[((ϕ^n)/(1-(ϕ^(2 n)))),{n,2,∞,2}])]

With[{ϕ=1/GoldenRatio},
(Sqrt[5])(NSum[((ϕ^(2n+1))/(1+(ϕ^(4n+2)))),{n,0,∞}]+NSum[((ϕ^(2n))/(1-(ϕ^(4 n)))),{n,∞}])]

V6

Sum[(1/(Fibonacci[n])),{n,∞}]

Sum[1/Simplify[FunctionExpand[Fibonacci[n]],n∈Integers],{n,∞}]

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

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

V7

SumConvergence[(1/(Fibonacci[n])),n]

Sum[(1/(Fibonacci[n])),n]//Timing

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

Sum[(1/(Fibonacci[n])),{n,∞}]//Timing

Sum[1/Simplify[FunctionExpand[Fibonacci[n]],n∈Integers],{n,∞}]//Timing

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

(Sqrt[5])With[{ϕ=1/GoldenRatio},NSum[((ϕ^(2k+1))/(1+(ϕ^(2 (2k+1))))),{k,0,∞}]+NSum[((ϕ^(2k))/(1-(ϕ^(4k)))),{k,∞}]]

TimeConstrained[(Sqrt[5])With[{ϕ=1/GoldenRatio},Sum[((ϕ^(2k+1))/(1+(ϕ^(2 (2k+1))))),{k,0,∞}]+Sum[((ϕ^(2k))/(1-(ϕ^(4k)))),{k,∞}]],60]//Timing

TimeConstrained[{Sum[((ϕ^n)/(1+(ϕ^(2 n)))),{n,1,∞,2}],Sum[((ϕ^n)/(1-(ϕ^(2 n)))),{n,2,∞,2}]},300]//Timing

With[{ϕ=1/GoldenRatio},
(Sqrt[5])(NSum[((ϕ^(2k+1))/(1+(ϕ^(2 (2k+1))))),{k,0,∞}]+NSum[((ϕ^(2k))/(1-(ϕ^(4k)))),{k,∞}])]

TimeConstrained[{Sum[((ϕ^(2k+1))/(1+(ϕ^(2 (2k+1))))),{k,0,∞}],Sum[((ϕ^(2k))/(1-(ϕ^(4k)))),{k,∞}]},300]//Timing

(Subscript[P, F]^((e)))==(∑_n^∞)(1/(Fibonacci[2 n]))==(Sqrt[5]) (∑_n^∞)(1/((ϕ^(2 n))-(ϕ^(-2n))))==(Sqrt[5]) (∑_n^∞)((ϕ^(2 n))/((ϕ^(4 n))-1))==(Sqrt[5]) (L((ϕ^(-2)))-L((ϕ^(-4))))==(((Sqrt[5]) (-(QPolyGamma3[0, 1, (ϕ^2)])+(QPolyGamma3[0, 1-((i π)/(2 log(ϕ))), (ϕ^2)])+i π))/(4 log(ϕ)))==(((Sqrt[5]) (2 (QPolyGamma3[0, 1, (ϕ^(-4))])-4 (QPolyGamma3[0, 1, (ϕ^(-2))])+log(5)))/(8 log(ϕ)))==1.5353705088362528`

NSum[1/Fibonacci[2n],{n,∞},WorkingPrecision->20]

Analytic

expr1=(Sqrt[5] (Log[5]+2 QPolyGamma[0,1,1/GoldenRatio^4]-4 QPolyGamma[0,1,1/GoldenRatio^2]))/(8 Log[GoldenRatio]);

expr2=Sqrt[5]/(4 Log[GoldenRatio]) (I π-QPolyGamma[0,1,GoldenRatio^2]+QPolyGamma[0,1-(I π)/(2 Log[GoldenRatio]),GoldenRatio^2]);

{expr1,expr2}//N

ByteCount/@{expr1,expr2}

V7

Sum[1/Fibonacci[2n],{n,∞}]//Timing

$Version

Sum[1/Fibonacci[2n],{n,∞}]//Timing

L-Function

FunctionExpand[GoldenRatio]

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

(Sqrt[5]) (∑_n^∞)((ϕ^(2 n))/((ϕ^(4 n))-1))

NSum[(((Sqrt[5]) (ϕ^(2 n)))/(-1+(ϕ^(4 n))))/.ϕ->GoldenRatio,{n,∞}]

(Sqrt[5])NSum[(( (ϕ^(2 n)))/((-1+(ϕ^(2 n)))(1+(ϕ^(2 n)))))/.ϕ->GoldenRatio,{n,∞}]

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

((Sqrt[5])/2)NSum[(1/(-1+(ϕ^(2n))))+(1/(1+(ϕ^(2n))))/.ϕ->GoldenRatio,{n,∞}]

(Sqrt[5])(L[(GoldenRatio^(-2))]-L[(GoldenRatio^(-4))])//N

(Sqrt[5])((((QPolyGamma[1,q]+Log[1-q])/(Log[q]))/.q->(GoldenRatio^(-2)))-(((QPolyGamma[1,q]+Log[1-q])/(Log[q]))/.q->(GoldenRatio^(-4))))//N

(Sqrt[5])(L[(GoldenRatio^2)]-L[(GoldenRatio^4)])//N

(∑_n^∞)((1/((ϕ^(-4 n))-1))-(1/((ϕ^(-2 n))-1)))

FullSimplify[(( (ϕ^(2 n)))/((-1+(ϕ^(2 n)))(1+(ϕ^(2 n)))))-((1/((ϕ^(-4n))-1))-(1/((ϕ^(-2n))-1)))/.ϕ->GoldenRatio,n∈Integers&&n>0]

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,1},{b,1},{k,4},{l,2}}]

(Sqrt[5])NSum[((1/((ϕ^(2 n))-1))-(1/((ϕ^(4 n))-1)))/.ϕ->GoldenRatio,{n,∞}]

(Sqrt[5])(L[(GoldenRatio^(-2))]-L[(GoldenRatio^(-4))])//N

((Sqrt[5])/2)Sum[(1/(-1+(ϕ^(2n))))+(1/(1+(ϕ^(2n))))/.ϕ->GoldenRatio,{n,∞}]//FullSimplify

N[%]

expr1=(Sqrt[5])(L[(GoldenRatio^(-2))]-L[(GoldenRatio^(-4))])

FullSimplify[expr1]

TraditionalForm[%]

(((Sqrt[5]) (Log[((-1+(GoldenRatio^4))/(((-1+(GoldenRatio^2)))^2))]+QPolyGamma[0,1,(1/(GoldenRatio^4))]-2 QPolyGamma[0,1,(1/(GoldenRatio^2))]))/(4 ArcCsch[2]))//FullSimplify

(((Sqrt[5]) (Log[5]+2 QPolyGamma[0,1,(1/(GoldenRatio^4))]-4 QPolyGamma[0,1,(1/(GoldenRatio^2))]))/(8 Log[GoldenRatio]))//N

expr2=((Sqrt[5])/(4 Log[GoldenRatio]))(I π-QPolyGamma[0,1,(GoldenRatio^2)]+QPolyGamma[0,1-((I π)/(2 Log[GoldenRatio])),(GoldenRatio^2)]);

{expr1,expr2}//N

FullSimplify[expr1-expr2]

N[%,1000]

PossibleZeroQ[%%]

q-Polygamma

Sum[1/Fibonacci[2n],n]//Timing

s=Sum[1/Fibonacci[2n],{n,k}]

{s/.k->10,Sum[1/Fibonacci[2n],{n,10}]}//N

Plot[s,{k,1,10}]

Limit[s,k->∞]

FullSimplify[%]

((1+Sqrt[5])/2)^2//Expand//Factor

s=((2^(-((2 I π)/(Log[(2/(3+(Sqrt[5])))])))) (((3+(Sqrt[5])))^((3 I π)/(Log[(2/(3+(Sqrt[5])))]))) (6460+2889 (Sqrt[5])) (-I π+QPolyGamma[0,1,(1/2) (3+(Sqrt[5]))]-QPolyGamma[0,1+((I π)/(Log[(2/(3+(Sqrt[5])))])),(1/2) (3+(Sqrt[5]))]))/((((9+4 (Sqrt[5])))^2) (-9 (2^((I π)/(Log[(2/(3+(Sqrt[5])))])))+4 (Sqrt[5]) (((3+(Sqrt[5])))^((I π)/(Log[(2/(3+(Sqrt[5])))])))) (Log[4]-2 Log[3+(Sqrt[5])]))/.{(3+Sqrt[5])->2GoldenRatio^2,(2/(3+(Sqrt[5])))->1/GoldenRatio^2}//FullSimplify

N[%]

Table[{n,((1+Sqrt[5])/2)^n//Expand//Factor},{n,20}]

FullSimplify[s/.{9+4 (Sqrt[5])->GoldenRatio^6}]

FullSimplify[(((6460+2889 (Sqrt[5])) )/((4 (GoldenRatio^18) ArcCsch[2])))]

((Sqrt[5])/(4 ArcCsch[2]))(I π-QPolyGamma[0,1,(GoldenRatio^2)]+QPolyGamma[0,1-((I π)/(2 ArcCsch[2])),(GoldenRatio^2)])//N

((Sqrt[5])/(4 Log[GoldenRatio]))(I π-QPolyGamma[0,1,(GoldenRatio^2)]+QPolyGamma[0,1-((I π)/(2 Log[GoldenRatio])),(GoldenRatio^2)])//N//Chop

NSum[1/Fibonacci[2n],{n,∞}]

(((Sqrt[5]) (-(QPolyGamma3[0, 1, (ϕ^2)])+(QPolyGamma3[0, 1-((I π)/(2 log(ϕ))), (ϕ^2)])+I π))/(4 log(ϕ)))==1.5353705088362528`

(Subscript[P, F]^((e)))==(∑_(n=0)^∞)(1/(Fibonacci[2 n+1]))==(Sqrt[5]) (∑_(n=0)^∞)((ϕ^(2 n+1))/((ϕ^(4 n+2))+1))==(((Sqrt[5]) ((EllipticTheta[2, 0, (1/(ϕ^2))])^2))/4)==(1/4) (Sqrt[5]) (((EllipticTheta[3, 0, (1/ϕ)])^2)-((EllipticTheta[3, 0, (1/(ϕ^2))])^2))==-(((Sqrt[5]) (π-i ((QPolyGamma3[0, (1/2)-((i π)/(4 log(ϕ))), (ϕ^2)])-(QPolyGamma3[0, (1/2)+((i π)/(4 log(ϕ))), (ϕ^2)]))))/(4 log(ϕ)))==1.82452

NSum[1/Fibonacci[2n+1],{n,0,∞}]

(1/4) (Sqrt[5]){((EllipticTheta[2,0,(GoldenRatio^(-2))])^2),((EllipticTheta[3,0,(GoldenRatio^(-1))])^2)-((EllipticTheta[3,0,(GoldenRatio^(-2))])^2)}//N

1/GoldenRatio^2//N

(1/2) (3-(Sqrt[5]))//N

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

Apart[(( ϕ)/((ϕ^(-2 n))+(ϕ^(2+2 n))))]

NSum[1/Fibonacci[2n+1],{n,∞}]

(Sqrt[5])NSum[((ϕ^(1+2 n))/(1+(ϕ^(2+4 n))))/.ϕ->GoldenRatio,{n,0,∞}]

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]

V6

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

V7

Sum[1/Fibonacci[2n+1],n]

s=Sum[1/Fibonacci[2n+1],{n,0,k}]

s/.k->99999.

(lim=TimeConstrained[Limit[s,k->∞],300])//Timing

(lim=TimeConstrained[Limit[s,k->∞],3600])//Timing

(lim=Limit[s,k->∞])//Timing

lim=(4 (((375125 π Log[(1/2) (3+(Sqrt[5]))])/(8 (Sqrt[2]) (((161+72 (Sqrt[5])))^2)))+((167761 (Sqrt[(5/2)]) π Log[(1/2) (3+(Sqrt[5]))])/(8 (((161+72 (Sqrt[5])))^2)))+(375125 I (2^((3/2)-((3 Log[2])/(2 ArcCsch[2]))-((3 (I π+Log[2]))/(2 ArcCsch[2]))+((3 I π)/(2 ArcCsch[2]+2 Log[2]-Log[4]))+((6 Log[2])/(2 ArcCsch[2]+2 Log[2]-Log[4])))) ArcCsch[2] QPolyGamma[0,-((-((I π)/2)+(1/2) (Log[2]-Log[3+(Sqrt[5])]))/(2 (ArcCsch[2]+Log[2])-Log[4])),(1/4) (((1+(Sqrt[5])))^2)])/((((3+(Sqrt[5])))^2) (((7+3 (Sqrt[5])))^2) (((9+4 (Sqrt[5])))^2))+(167761 I (2^((3/2)-((3 Log[2])/(2 ArcCsch[2]))-((3 (I π+Log[2]))/(2 ArcCsch[2]))+((3 I π)/(2 ArcCsch[2]+2 Log[2]-Log[4]))+((6 Log[2])/(2 ArcCsch[2]+2 Log[2]-Log[4])))) (Sqrt[5]) ArcCsch[2] QPolyGamma[0,-((-((I π)/2)+(1/2) (Log[2]-Log[3+(Sqrt[5])]))/(2 (ArcCsch[2]+Log[2])-Log[4])),(1/4) (((1+(Sqrt[5])))^2)])/((((3+(Sqrt[5])))^2) (((7+3 (Sqrt[5])))^2) (((9+4 (Sqrt[5])))^2))+(375125 I (2^((3/2)-((3 Log[2])/(2 ArcCsch[2]))-((3 (I π+Log[2]))/(2 ArcCsch[2]))+((3 I π)/(2 ArcCsch[2]+2 Log[2]-Log[4]))+((6 Log[2])/(2 ArcCsch[2]+2 Log[2]-Log[4])))) Log[2] QPolyGamma[0,-((-((I π)/2)+(1/2) (Log[2]-Log[3+(Sqrt[5])]))/(2 (ArcCsch[2]+Log[2])-Log[4])),(1/4) (((1+(Sqrt[5])))^2)])/((((3+(Sqrt[5])))^2) (((7+3 (Sqrt[5])))^2) (((9+4 (Sqrt[5])))^2))+(167761 I (2^((3/2)-((3 Log[2])/(2 ArcCsch[2]))-((3 (I π+Log[2]))/(2 ArcCsch[2]))+((3 I π)/(2 ArcCsch[2]+2 Log[2]-Log[4]))+((6 Log[2])/(2 ArcCsch[2]+2 Log[2]-Log[4])))) (Sqrt[5]) Log[2] QPolyGamma[0,-((-((I π)/2)+(1/2) (Log[2]-Log[3+(Sqrt[5])]))/(2 (ArcCsch[2]+Log[2])-Log[4])),(1/4) (((1+(Sqrt[5])))^2)])/((((3+(Sqrt[5])))^2) (((7+3 (Sqrt[5])))^2) (((9+4 (Sqrt[5])))^2))-(375125 I (2^((1/2)-((3 Log[2])/(2 ArcCsch[2]))-((3 (I π+Log[2]))/(2 ArcCsch[2]))+((3 I π)/(2 ArcCsch[2]+2 Log[2]-Log[4]))+((6 Log[2])/(2 ArcCsch[2]+2 Log[2]-Log[4])))) Log[4] QPolyGamma[0,-((-((I π)/2)+(1/2) (Log[2]-Log[3+(Sqrt[5])]))/(2 (ArcCsch[2]+Log[2])-Log[4])),(1/4) (((1+(Sqrt[5])))^2)])/((((3+(Sqrt[5])))^2) (((7+3 (Sqrt[5])))^2) (((9+4 (Sqrt[5])))^2))-(167761 I (2^((1/2)-((3 Log[2])/(2 ArcCsch[2]))-((3 (I π+Log[2]))/(2 ArcCsch[2]))+((3 I π)/(2 ArcCsch[2]+2 Log[2]-Log[4]))+((6 Log[2])/(2 ArcCsch[2]+2 Log[2]-Log[4])))) (Sqrt[5]) Log[4] QPolyGamma[0,-((-((I π)/2)+(1/2) (Log[2]-Log[3+(Sqrt[5])]))/(2 (ArcCsch[2]+Log[2])-Log[4])),(1/4) (((1+(Sqrt[5])))^2)])/((((3+(Sqrt[5])))^2) (((7+3 (Sqrt[5])))^2) (((9+4 (Sqrt[5])))^2))-(375125 I (2^((3/2)-((3 Log[2])/(2 ArcCsch[2]))-((3 (I π+Log[2]))/(2 ArcCsch[2]))+((3 I π)/(2 ArcCsch[2]+2 Log[2]-Log[4]))+((6 Log[2])/(2 ArcCsch[2]+2 Log[2]-Log[4])))) ArcCsch[2] QPolyGamma[0,-((((I π)/2)+(1/2) (Log[2]-Log[3+(Sqrt[5])]))/(2 (ArcCsch[2]+Log[2])-Log[4])),(1/4) (((1+(Sqrt[5])))^2)])/((((3+(Sqrt[5])))^2) (((7+3 (Sqrt[5])))^2) (((9+4 (Sqrt[5])))^2))-(167761 I (2^((3/2)-((3 Log[2])/(2 ArcCsch[2]))-((3 (I π+Log[2]))/(2 ArcCsch[2]))+((3 I π)/(2 ArcCsch[2]+2 Log[2]-Log[4]))+((6 Log[2])/(2 ArcCsch[2]+2 Log[2]-Log[4])))) (Sqrt[5]) ArcCsch[2] QPolyGamma[0,-((((I π)/2)+(1/2) (Log[2]-Log[3+(Sqrt[5])]))/(2 (ArcCsch[2]+Log[2])-Log[4])),(1/4) (((1+(Sqrt[5])))^2)])/((((3+(Sqrt[5])))^2) (((7+3 (Sqrt[5])))^2) (((9+4 (Sqrt[5])))^2))-(375125 I (2^((3/2)-((3 Log[2])/(2 ArcCsch[2]))-((3 (I π+Log[2]))/(2 ArcCsch[2]))+((3 I π)/(2 ArcCsch[2]+2 Log[2]-Log[4]))+((6 Log[2])/(2 ArcCsch[2]+2 Log[2]-Log[4])))) Log[2] QPolyGamma[0,-((((I π)/2)+(1/2) (Log[2]-Log[3+(Sqrt[5])]))/(2 (ArcCsch[2]+Log[2])-Log[4])),(1/4) (((1+(Sqrt[5])))^2)])/((((3+(Sqrt[5])))^2) (((7+3 (Sqrt[5])))^2) (((9+4 (Sqrt[5])))^2))-(167761 I (2^((3/2)-((3 Log[2])/(2 ArcCsch[2]))-((3 (I π+Log[2]))/(2 ArcCsch[2]))+((3 I π)/(2 ArcCsch[2]+2 Log[2]-Log[4]))+((6 Log[2])/(2 ArcCsch[2]+2 Log[2]-Log[4])))) (Sqrt[5]) Log[2] QPolyGamma[0,-((((I π)/2)+(1/2) (Log[2]-Log[3+(Sqrt[5])]))/(2 (ArcCsch[2]+Log[2])-Log[4])),(1/4) (((1+(Sqrt[5])))^2)])/((((3+(Sqrt[5])))^2) (((7+3 (Sqrt[5])))^2) (((9+4 (Sqrt[5])))^2))+(375125 I (2^((1/2)-((3 Log[2])/(2 ArcCsch[2]))-((3 (I π+Log[2]))/(2 ArcCsch[2]))+((3 I π)/(2 ArcCsch[2]+2 Log[2]-Log[4]))+((6 Log[2])/(2 ArcCsch[2]+2 Log[2]-Log[4])))) Log[4] QPolyGamma[0,-((((I π)/2)+(1/2) (Log[2]-Log[3+(Sqrt[5])]))/(2 (ArcCsch[2]+Log[2])-Log[4])),(1/4) (((1+(Sqrt[5])))^2)])/((((3+(Sqrt[5])))^2) (((7+3 (Sqrt[5])))^2) (((9+4 (Sqrt[5])))^2))+(167761 I (2^((1/2)-((3 Log[2])/(2 ArcCsch[2]))-((3 (I π+Log[2]))/(2 ArcCsch[2]))+((3 I π)/(2 ArcCsch[2]+2 Log[2]-Log[4]))+((6 Log[2])/(2 ArcCsch[2]+2 Log[2]-Log[4])))) (Sqrt[5]) Log[4] QPolyGamma[0,-((((I π)/2)+(1/2) (Log[2]-Log[3+(Sqrt[5])]))/(2 (ArcCsch[2]+Log[2])-Log[4])),(1/4) (((1+(Sqrt[5])))^2)])/((((3+(Sqrt[5])))^2) (((7+3 (Sqrt[5])))^2) (((9+4 (Sqrt[5])))^2))))/((Sqrt[3+(Sqrt[5])]) (2 ArcCsch[2]+2 Log[2]-Log[4]) (Log[2]-Log[3+(Sqrt[5])]));

N[lim]

FullSimplify[lim]//Timing

lim=((Sqrt[5]) (π ArcCosh[(3/2)]-2 I ArcCsch[2] (QPolyGamma[0,((-I π+ArcCosh[(3/2)])/(4 ArcCsch[2])),(1/2) (3+(Sqrt[5]))]-QPolyGamma[0,((I π+ArcCosh[(3/2)])/(4 ArcCsch[2])),(1/2) (3+(Sqrt[5]))])))/(4 ArcCsch[2] Log[(1/2) (3-(Sqrt[5]))]);

N[lim]

Factor//@TrigToExp[lim]

((2(3+(Sqrt[5])))/(Expand[(3+(Sqrt[5]))(3-(Sqrt[5]))]))

Log[(1/2) (3-(Sqrt[5]))] Log[(1/2) (1+(Sqrt[5]))])

Factor//@TrigToExp[lim]/.{(3+(Sqrt[5]))->2GoldenRatio^2,(3-(Sqrt[5]))->2/GoldenRatio^2,(1+(Sqrt[5]))->2GoldenRatio}

FullSimplify[%18]/. ArcCsch[2]->Log[GoldenRatio]

N[%]

TraditionalForm[%27]

Sum[1/Fibonacci[2n+1],{n,0,∞}]//Timing

NSum[1/Fibonacci[2n+1],{n,0,∞}]

Sum[1/Fibonacci[2n+1],{n,0,∞}]//Timing

expr1=(1/4) (Sqrt[5]) ((EllipticTheta[2,0,(1/2) (3-(Sqrt[5]))])^2);
expr2=-(1/(4 Log[GoldenRatio]))(Sqrt[5]) (π-I (QPolyGamma[0,(1/2)-((I π)/(4 Log[GoldenRatio])),(GoldenRatio^2)]-QPolyGamma[0,(1/2)+((I π)/(4 Log[GoldenRatio])),(GoldenRatio^2)]));

N[{expr1,expr2}]

FullSimplify[expr1-expr2]

PossibleZeroQ[expr1-expr2]

N[expr1-expr2,1000]

N[%]

FullSimplify[FunctionExpand[(1/(Fibonacci[1+2 n]))],n∈Integers&&n>0]

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

FullSimplify[(((((2 (1+(Sqrt[5]))))^(2 n)) (5+(Sqrt[5])))/((2^(1+4 n))+(((1+(Sqrt[5])))^(4 n)) (3+(Sqrt[5]))))/.(1+Sqrt[5]):>2/Φ,Φ>0]

With[{q=1/GoldenRatio^2},(1/2)(5+(Sqrt[5]))NSum[(( (q^n))/(1/q+(q^(2n)))),{n,∞}]]

With[{q=1/GoldenRatio^2},(1/2)(5+(Sqrt[5]))NSum[(( (q^(n+1)))/(1+(q^(2n+1)))),{n,∞}]]

With[{q=1/GoldenRatio^2},(1/2)(5+(Sqrt[5]))NSum[(( (q^(n+1)))/(1+(q^(2n+1)))),{n,∞}]]

With[{q=1/GoldenRatio^2},(((1/2)(5+(Sqrt[5])))/(q^(-1/2)))NSum[(( 1)/((q^(-(n+1/2)))+(q^(n+1/2)))),{n,∞}]]