In mathematical analysis, the Foias constant is a real number named after Ciprian Foias.
It is defined in the following way: for every real number x1 > 0, there is a sequence defined by the recurrence relation
\( {\displaystyle x_{n+1}=\left(1+{\frac {1}{x_{n}}}\right)^{n}} \)
for n = 1, 2, 3, .... The Foias constant is the unique choice α such that if x1 = α then the sequence diverges to infinity.[1] Numerically, it is
\( {\displaystyle \alpha =1.187452351126501\ldots } \) OEIS: A085848.
No closed form for the constant is known.
When x1 = α then we have the limit:
\( \lim _{{n\to \infty }}x_{n}{\frac {\log n}n}=1, \)
where "log" denotes the natural logarithm. Consequently, one has by the prime number theorem that in this case
l \( \lim _{{n\to \infty }}{\frac {x_{n}}{\pi (n)}}=1, \)
where π is the prime-counting function.[1]
See also
Mathematical constant
Notes and references
Ewing, J. and Foias, C. "An Interesting Serendipitous Real Number." In Finite versus Infinite: Contributions to an Eternal Dilemma (Ed. C. Caluse and G. Păun). London: Springer-Verlag, pp. 119–126, 2000.
Weisstein, Eric W. "Foias Constant". MathWorld.
S. R. Finch (2003). Mathematical Constants. Cambridge University Press. p. 430. ISBN 0-521-818-052. "Foias constant."
Sloane, N. J. A. (ed.). "Sequence A085848 (Decimal expansion of Foias's Constant)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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