Reciprocal Fibonacci constant
Mathematical constant From Wikipedia, the free encyclopedia
The reciprocal Fibonacci constant ψ is the sum of the reciprocals of the Fibonacci numbers:
Because the ratio of successive terms tends to the reciprocal of the golden ratio, which is less than 1, the ratio test shows that the sum converges.
The value of ψ is approximately
(sequence A079586 in the OEIS).
With k terms, the series gives O(k) digits of accuracy. Bill Gosper derived an accelerated series which provides O(k 2) digits.[1] ψ is irrational, as was conjectured by Paul Erdős, Ronald Graham, and Leonard Carlitz, and proved in 1989 by Richard André-Jeannin.[2]
Its simple continued fraction representation is:
Generalization and related constants
Summarize
Perspective
In analogy to the Riemann zeta function, define the Fibonacci zeta function as for complex number s with Re(s) > 0, and its analytic continuation elsewhere. Particularly the given function equals ψ when s = 1.[3]
It was shown that:
- The value of ζF (2s) is transcendental for any positive integer s, which is similar to the case of even-index Riemann zeta-constants ζ(2s).[3][4]
- The constants ζF (2), ζF (4) and ζF (6) are algebraically independent.[3][4]
- Except for ζF (1) which was proved to be irrational, the number-theoretic properties of ζF (2s + 1) (whenever s is a non-negative integer) are mostly unknown.[3]
See also
References
External links
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