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

Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...

s: :

\psi = \sum_^ \frac = \frac +  \frac + \frac + \frac + \frac + \frac + \frac + \frac + \cdots.

The ratio of successive terms in this sum tends to the reciprocal of the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

. Since this is less than 1, the

ratio test In mathematics, the ratio test is a test (or "criterion") for the convergence of a series :\sum_^\infty a_n, where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert ...

shows that the sum converges. The value of ψ is known to be approximately :

\psi = 3.359885666243177553172011302918927179688905133732\dots .

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. ψ is known to be

irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. ...

; this property was conjectured by Paul Erdős,

Ronald Graham Ronald Lewis Graham (October 31, 1935July 6, 2020) was an American mathematician credited by the American Mathematical Society as "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years". He ...

, and

Leonard Carlitz Leonard Carlitz (December 26, 1907 – September 17, 1999) was an American mathematician. Carlitz supervised 44 doctorates at Duke University and published over 770 papers. Chronology * 1907 Born Philadelphia, PA, USA * 1927 BA, Universi ...

, and proved in 1989 by Richard André-Jeannin. The

continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integ ...

representation of the constant is: :

\psi =;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 \!\, .

References

External links

* Mathematical constants Fibonacci numbers Irrational numbers {{Math-stub

See also

References

External links