Reciprocal Fibonacci constant

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

\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

\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 $O(k 2)$ 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]