The reciprocal Fibonacci constant, or
ψ, is defined as the sum of the
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
s of the
Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a 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 the sequence from ...
s:
:
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 a ...
shows that the sum converges.
The value of ψ is known to be approximately
:
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. T ...
; this property was conjectured by
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
,
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, University ...
, and proved in 1989 by
Richard André-Jeannin
Richard is a male given name. It originates, via Old French, from Old Frankish and is a compound of the words descending from Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'stron ...
.
The
continued fraction
In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...
representation of the constant is:
:
See also
*
List of sums of reciprocals
In mathematics and especially number theory, the sum of reciprocals generally is computed for the reciprocals of some or all of the positive integers (counting numbers)—that is, it is generally the sum of unit fractions. If infinitely many n ...
References
External links
*
Mathematical constants
Fibonacci numbers
Irrational numbers
{{Math-stub