Foias constant

{{Short description|Mathematical constant}}

File:Foias constant sequence.png is used.]]

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

: $x\_\{n+1\}\; =\; \backslash left(\; 1\; +\; \backslash frac\{1\}\{x\_n\}\; \backslash right)^n$

for n = 1, 2, 3, .... The Foias constant is the unique choice α such that if x₁ = α then the sequence diverges to infinity. For all other values of x₁, the sequence is divergent as well, but it has two accumulation points: 1 and infinity.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. Numerically, it is

: $\backslash alpha\; =\; 1.187452351126501\backslash ldots$.{{Cite OEIS|sequencenumber=A085848 |name=Decimal expansion of Foias's Constant}}

No closed form for the constant is known.

When x₁ = α then the growth rate of the sequence (x_n) is given by the limit

: $\backslash lim\_\{n\backslash to\backslash infty\}\; x\_n\; \backslash frac\{\backslash log\; n\}n\; =\; 1,$

where "log" denotes the natural logarithm.

The same methods used in the proof of the uniqueness of the Foias constant may also be applied to other similar recursive sequences.{{citation|last = Anghel|first = Nicolae| title = Foias numbers|journal = An. Ştiinţ. Univ. "Ovidius" Constanţa Ser. Mat.|volume = 26| year = 2018| issue = 3| pages = 21–28 | doi = 10.2478/auom-2018-0030|s2cid = 195842026| url = https://digital.library.unt.edu/ark:/67531/metadc1705461/m2/1/high_res_d/18440835-Analele_Universitatii_Ovidius.pdf}}