View source for Reciprocal Fibonacci constant

The '''reciprocal Fibonacci constant''', or [[ψ]], is defined as the sum of the [[Reciprocal (mathematics)|reciprocal]]s of the [[Fibonacci number]]s:

:<math>\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.</math>

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

:<math>\psi = 3.359885666243177553172011302918927179688905133732\dots .</math> {{OEIS|A079586}}

[[Bill Gosper|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 [[Series acceleration|accelerated series]] provides O(''k''<sup>2</sup>) digits.<ref>{{citation
 | last = Gosper
 | first = William R.
 | authorlink = Bill Gosper
 | year = 1974
 | title = Acceleration of Series
 | page = 66
 | url = http://dspace.mit.edu/handle/1721.1/6088
 | publisher = Artificial Intelligence Memo #304, Artificial Intelligence Laboratory, [[Massachusetts Institute of Technology]]}}.</ref>
ψ is known to be [[irrational number|irrational]]; this property was conjectured by [[Paul Erdős]], [[Ronald Graham]], and [[Leonard Carlitz]], and proved in 1989 by [[Richard André-Jeannin]].<ref>{{citation
 | last = André-Jeannin
 | first = Richard
 | title = Irrationalité de la somme des inverses de certaines suites récurrentes
 | url = http://gallica.bnf.fr/ark:/12148/bpt6k5686125p/f9.image
 | journal = [[Comptes Rendus de l'Académie des Sciences, Série I]]
 | volume = 308
 | year = 1989
 | issue = 19
 | pages = 539–541
 |mr=0999451}}</ref>

The [[continued fraction]] representation of the constant is:

: <math>\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 ] \!\, . </math> {{OEIS|A079587}}

==See also==

*[[List of sums of reciprocals]]

==References==
<references/>

==External links==
*{{MathWorld|title = Reciprocal Fibonacci Constant | urlname = ReciprocalFibonacciConstant}}

[[Category:Mathematical constants]]
[[Category:Fibonacci numbers]]
[[Category:Irrational numbers]]


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