Reciprocal Fibonacci constant

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

:$psi\; approx\; 3.359885666243177553172011302918927179688905133731\; dots\; .$ [OEIS|id=A079586]

No closed formula for ψ is known, but Gosper describes an algorithm for fast numerical approximation of its value. [The reciprocal Fibonacci series itself provides O("k") digits of accuracy for "k" terms of expansion, while Gosper's accelerated series provides O("k"²) digits. citation
last = Gosper
first = William R.
authorlink = Bill Gosper
year = 1974
title = Acceleration of Series
pages = p.66
url = http://dspace.mit.edu/handle/1721.1/6088
publisher = Artificial Intelligence Memo #304, Artificial Intelligence Laboratory, Massachusetts Institute of Technology.] 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. [citation
last = André-Jeannin
first = Richard
title = Irrationalité de la somme des inverses de certaines suites récurrentes
journal = C. R. Acad. Sci. Paris Sér. I Math.
volume = 308
year = 1989
issue = 19
pages = 539–541
id = MathSciNet | id = 0999451]

The continued fraction representation of the constant is:

: $psi\; =\; [3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,2,4,8,6,30,50,1,6,3,3,2,7,2,3,1,3,2,\; cdots\; ]\; !,\; .$ [OEIS|id=A079587]

References

External links

*MathWorld|title = Reciprocal Fibonacci Constant | urlname = ReciprocalFibonacciConstant