Reciprocal Fibonacci constant

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

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

The ratio of successive terms in this sum tends to the reciprocal of the golden ratio. Since this 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).

Bill Gosper describes an algorithm for its fast numerical approximation. The reciprocal Fibonacci series provides O(k) digits of accuracy for k terms, while Gosper's accelerated series provides O(k²) digits. ψ is known to be irrational; this property was conjectured by Paul Erdős, Ronald Graham, and Leonard Carlitz, and proven in 1989 by Richard André-Jeannin.

Its 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).

See also

• List of sums of reciprocals

External links

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