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
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:
![{\displaystyle \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 ]\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3c66dc6fb55a42af1ae437279f1720b45fcbc49)
See also
[edit]References
[edit]- ^ Gosper, William R. (1974), Acceleration of Series, Artificial Intelligence Memo #304, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, p. 66, hdl:1721.1/6088.
- ^ 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
External links
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Well, that’s interesting to know that Psilotum nudum are known as whisk ferns. Psilotum nudum is the commoner species of the two. While the P. flaccidum is a rare species and is found in the tropical islands. Both the species are usually epiphytic in habit and grow upon tree ferns. These species may also be terrestrial and grow in humus or in the crevices of the rocks.
View the detailed Guide of Psilotum nudum: Detailed Study Of Psilotum Nudum (Whisk Fern), Classification, Anatomy, Reproduction