The reciprocal Fibonacci constant ψ is the sum of the reciprocals of the Fibonacci numbers:
ψ =
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:
(sequence A079587 in the OEIS).
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.
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