Reciprocal Fibonacci constant

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 continued fraction representation is:

(sequence A079587 in the OEIS).

See also

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References

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  1. ^ Gosper, William R. (1974), Acceleration of Series, Artificial Intelligence Memo #304, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, p. 66, hdl:1721.1/6088.
  2. ^ André-Jeannin, Richard (1989), "Irrationalité de la somme des inverses de certaines suites récurrentes", Comptes Rendus de l'Académie des Sciences, Série I, 308 (19): 539–541, MR 0999451
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