Reciprocal Lucas Constant
Closed forms are known for the sums of reciprocals of even-indexed Lucas numbers
P_L^((e)) = [画像:sum_(n=1)^(infty)1/(L_(2n))]
(1)
![画像:-1/(4lnphi){pi+i[psi_(phi^2)(1+(ipi)/(4lnphi))-psi_(phi^2)(1-(ipi)/(4lnphi))]}](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ReciprocalLucasConstant/Inline9.svg){kind=link}
= 1/4[theta_3^2(phi^(-2))-1]
(4)
= 0.566177675...
(5)
(OEIS A153415), where phi is the golden ratio, psi_q^((n))(z) is a q-polygamma function, and theta_n(q) is a Jacobi theta function, and odd-indexed Lucas numbers
P_L^((o)) = [画像:sum_(n=0)^(infty)1/(L_(2n+1))]
(6)
= L(phi^(-4))-2L(phi^(-2))+L(phi^(-1))
(8)
![画像:1/(4lnphi)[7lnphi-ln(phi^2+1)-4psi_(phi^(-1))(1)+4psi_(phi^(-2))(1)-psi_(phi^(-4))(1)]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ReciprocalLucasConstant/Inline30.svg){kind=link}
![画像:1/(4lnphi)[psi_(phi^2)(1/2-(ipi)/(2lnphi))-psi_(phi^2)(1/2)+ipi]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ReciprocalLucasConstant/Inline33.svg){kind=link}
= 1.39668...
(11)
(OEIS A153416), where L(beta) is a Lambert series (Borwein and Borwein 1987, pp. 91-92). This gives the reciprocal Lucas constant as
P_L = [画像:sum_(n=1)^(infty)1/(L_n)]
(12)
= [画像:sum_(n=1)^(infty)(F_n)/(F_(2n))]
(14)
= P_L^((e))+P_L^((o))
(15)
= 1.96285817...
(16)
(OEIS A093540), where phi is the golden ratio and F_n is a Fibonacci number.
Borwein and Borwein (1987, pp. 94-101) give a number of related beautiful formulas.
See also
Lucas Number, Lambert Series, q-Polygamma Function, Reciprocal Fibonacci ConstantExplore with Wolfram|Alpha
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References
Borwein, J. M. and Borwein, P. B. "Evaluation of Sums of Reciprocals of Fibonacci Sequences." §3.7 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 91-101, 1987.Sloane, N. J. A. Sequences A093540, A153415, and A153416 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Reciprocal Lucas ConstantCite this as:
Weisstein, Eric W. "Reciprocal Lucas Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ReciprocalLucasConstant.html