Reciprocal Fibonacci constant

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

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

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 known to be approximately

$\psi \approx 3.359885666243177553172011302918927179688905133731 \dots .$ ^[1]

No closed formula for ψ is known, but Gosper describes an algorithm for fast numerical approximation of its value.^[2] ψ is known to be irrational; this property was conjectured by Paul Erdős, Ronald Graham, and Leonard Carlitz, and proved in 1989 by Richard André-Jeannin.^[3]

The continued fraction representation of the constant is:

$\psi =[3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,2,4,8,6,30,50,1,6,3,3,2,7,2,3,1,3,2, \cdots ] \!\, .$ ^[4]

[edit] References

^ (sequence A079586 in OEIS)
^ The reciprocal Fibonacci series itself provides O(k) digits of accuracy for k terms of expansion, while Gosper's accelerated series provides O(k²) digits. Gosper, William R. (1974), Acceleration of Series, Artificial Intelligence Memo #304, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, pp. p.66, <http://dspace.mit.edu/handle/1721.1/6088> .
^ André-Jeannin, Richard (1989), “Irrationalité de la somme des inverses de certaines suites récurrentes”, C. R. Acad. Sci. Paris Sér. I Math. 308 (19): 539–541, MR 0999451
^ (sequence A079587 in OEIS)