The reciprocal Fibonacci constant is the sum of the
reciprocals of the
Fibonacci number
In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...
s:
Because the ratio of successive terms 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 summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if
\fr ...
, which is less than 1, the
ratio test
In mathematics, the ratio test is a convergence tests, test (or "criterion") for the convergent series, convergence of a series (mathematics), series
:\sum_^\infty a_n,
where each term is a real number, real or complex number and is nonzero wh ...
shows that the sum
converges.
The value of is approximately
.
With terms, the series gives digits of accuracy.
Bill Gosper derived an
accelerated series which provides digits.
is
irrational
Irrationality is cognition, thinking, talking, or acting without rationality.
Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
, as was
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
d by
Paul Erdős,
Ronald Graham, and
Leonard Carlitz, and
proved in 1989 by
Richard André-Jeannin.
Its
simple continued fraction
A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence \ of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fr ...
representation is:
.
Generalization and related constants
In analogy to the
Riemann zeta function, define the Fibonacci zeta function as
for
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
with , and its
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ne ...
elsewhere. Particularly the given function equals when .
It was shown that:
* The value of is transcendental for any positive integer , which is similar to the case of even-index
Riemann zeta-constants .
* The constants , and are
algebraically independent
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non- trivial polynomial equation with coefficients in K.
In particular, a one element set \ is algebraically i ...
.
* Except for which was proved to be irrational, the number-theoretic properties of (whenever s is a non-
negative integer) are mostly unknown.
See also
*
List of sums of reciprocals
*
List of mathematical constants
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. For e ...
References
External links
*
Mathematical constants
Fibonacci numbers
Irrational numbers
{{Math-stub