Closed forms are known for the sums of reciprocals of even-indexed Fibonacci
numbers
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(OEIS A153386; Knopp 1990, Ch. 8, Ex. 114; Paszkowski 1997; Horadam 1988; Finch 2003, p. 358; E. Weisstein, Jan. 1,
2009; Arndt 2012), where is the golden ratio,
is a q-polygamma function, and
is a Lambert series (Borwein and Borwein 1987,
pp. 91 and 95) and odd-indexed Fibonacci numbers
(8)
(9)
(10)
(11)
(12)
(13)
(OEIS A153387; Landau 1899; Borwein and Borwein 1997, p. 94; E. Weisstein, Jan. 1, 2009; Arndt 2012), where is a Jacobi
elliptic function. Together, these give a closed form for the reciprocal Fibonacci
constant of
(14)
(15)
(16)
(17)
(18)
(OEIS A079586; Horadam 1988; Griffin 1992; Zhao 1999; Finch 2003, p. 358). The question of the irrationality of was formally raised by Paul Erdős and this sum was
proved to be irrational by André-Jeannin (1989).
Borwein and Borwein (1987, pp. 94-101) give a number of beautiful related formulas.
André-Jeannin, R. "Irrationalité de la somme des inverses de certaines suites récurrentes." C. R. Acad. Sci.
Paris Sér. I Math.308, 539-541, 1989.Apéry,
F. "Roger Apéry, 1916-1994: A Radical Mathematician." Math. Intell.18,
No. 2, 54-61, 1996.Arndt, J. "On Computing the Generalized
Lambert Series." 24 Jun 2012. http://arxiv.org/abs/1202.6525.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.Bundschuh, P. and Väänänen,
K. "Arithmetical Investigations of a Certain Infinite Product." Compos.
Math.91, 175-199, 1994.Duverney, D. "Irrationalité
de la somme des inverses de la suite de Fibonacci." Elem. Math.52,
31-36, 1997.Finch, S. R. Mathematical
Constants. Cambridge, England: Cambridge University Press, p. 358, 2003.Griffin,
P. "Acceleration of the Sum of Fibonacci Reciprocals." Fib. Quart.30,
179-181, 1992.Horadam, A. F. "Elliptic Functions and Lambert
Series in the Summation of Reciprocals in Certain Recurrence-Generated Sequences."
Fib. Quart.26, 98-114, 1988.Knopp, K. Theory
and Application of Infinite Series. New York: Dover, 1990.Landau,
E. "Sur la série des inverse de nombres de Fibonacci." Bull.
Soc. Math. France27, 298-300, 1899.Paszkowski, S. "Fast
Convergent Quasipower Series for Some Elementary and Special Functions." Comput.
Math. Appl.33, 181-191, 1997.Prévost, M. "On
the Irrationality of ." J. Number Th.73,
139-161, 1998.Michon, G. P. "Final Answers: Numerical Constants."
http://home.att.net/~numericana/answer/constants.htm#prevost.Sloane,
N. J. A. Sequences A079586, A153386,
and A153387 in "The On-Line Encyclopedia
of Integer Sequences."Zhao, F.-Z. "Notes of Reciprocal Series
Related to the Fibonacci and Lucas Numbers." Fib. Quart.37, 254-257,
1999.
![sqrt(5)[L(phi^(-2))-L(phi^(-4))]](/images/equations/ReciprocalFibonacciConstant/Inline12.svg)
![(sqrt(5))/(8lnphi)[ln5+2psi_(phi^(-4))(1)-4psi_(phi^(-2))(1)]](/images/equations/ReciprocalFibonacciConstant/Inline15.svg)
![(sqrt(5))/(4lnphi)[psi_(phi^2)(1-(ipi)/(2lnphi))-psi_(phi^2)(1)+ipi]](/images/equations/ReciprocalFibonacciConstant/Inline18.svg)
is the golden ratio, 
is a q-polygamma function, and 
is a Lambert series (Borwein and Borwein 1987,
pp. 91 and 95) and odd-indexed Fibonacci numbers
![-(sqrt(5))/(4lnphi){pi-i[psi_(phi^2)(1/2-(ipi)/(4lnphi))-psi_(phi^2)(1/2+(ipi)/(4lnphi))]}](/images/equations/ReciprocalFibonacciConstant/Inline33.svg)
![1/4sqrt(5)[theta_3^2(phi^(-1))-theta_3^2(phi^(-2))]](/images/equations/ReciprocalFibonacciConstant/Inline39.svg)
is a Jacobi
elliptic function. Together, these give a closed form for the reciprocal Fibonacci
constant of
![(sqrt(5))/4[(ln5+2psi_(phi^(-4))(1)-4psi_(phi^(-2))(1))/(2lnphi)+theta_2^2(phi^(-2))]](/images/equations/ReciprocalFibonacciConstant/Inline55.svg)
was formally raised by Paul Erdős and this sum was
proved to be irrational by André-Jeannin (1989).
." J. Number Th. 73,
139-161, 1998.Michon, G. P. "Final Answers: Numerical Constants."