Reciprocal Fibonacci constant

Template:Short description The reciprocal Fibonacci constant Template:Mvar is the sum of the reciprocals of the Fibonacci numbers:

$$\psi =\sum _{k=1}^{\mathrm{\infty }}\frac{1}{F_k}=\frac{1}{1}+\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{8}+\frac{1}{13}+\frac{1}{21}+\cdots .$$

Because the ratio of successive terms tends to the reciprocal of the golden ratio, which is less than 1, the ratio test shows that the sum converges.

The value of Template:Mvar is approximately

$$\psi =3.359885666243177553172011302918927179688905133732\dots$$ (sequence A079586 in the OEIS).

With Template:Mvar terms, the series gives O(k) digits of accuracy. Bill Gosper derived an accelerated series which provides O(k²) digits. Template:Mvar is irrational, as was conjectured by Paul Erdős, Ronald Graham, and Leonard Carlitz, and proved in 1989 by Richard André-Jeannin.

Its simple continued fraction representation is:

$$\psi =[3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,2,4,8,6,30,50,1,6,3,3,2,7,2,3,1,3,2,\dots ]$$ (sequence A079587 in the OEIS).

Generalization and related constants

In analogy to the Riemann zeta function, define the Fibonacci zeta function as $${\zeta }_F(s)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{(F_n{)}^s}=\frac{1}{1^s}+\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{5^s}+\frac{1}{8^s}+\cdots$$ for complex number Template:Mvar with Re(s) > 0, and its analytic continuation elsewhere. Particularly the given function equals Template:Mvar when s = 1.

It was shown that:

• The value of ζ_F(2s) is transcendental for any positive integer Template:Mvar, which is similar to the case of even-index Riemann zeta-constants ζ(2s).
• The constants ζ_F(2), ζ_F(4) and ζ_F(6) are algebraically independent.
• Except for ζ_F(1) which was proved to be irrational, the number-theoretic properties of ζ_F(2s + 1) (whenever s is a non-negative integer) are mostly unknown.

See also

• List of sums of reciprocals
• List of mathematical constants