Reciprocal Fibonacci constant

The reciprocal Fibonacci constant ψ is the sum of the reciprocals of the Fibonacci numbers:

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

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 ψ is approximately

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

With k terms, the series gives O(k) digits of accuracy. Bill Gosper derived an accelerated series which provides O(k²) digits.^[1] ψ is irrational, as was conjectured by Paul Erdős, Ronald Graham, and Leonard Carlitz, and proved in 1989 by Richard André-Jeannin.^[2]

Its continued fraction representation is:

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

• List of sums of reciprocals

• Weisstein, Eric W. "Reciprocal Fibonacci Constant". MathWorld.