Foias constant

{\displaystyle x_{n+1}=(1+1/x_{n})^{n}} — Evolution of the sequence $x_{n+1}=(1+1/x_{n})^{n}$ for several values of $x_{1}$ , around the Foias constant $\alpha$ . Evolution for $x_{1}=\alpha$ is in green. Other initial values lead to two accumulation points, 1 and $\infty$ . A logarithmic scale 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}=\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. For all other values of x₁, the sequence is divergent as well, but it has two accumulation points: 1 and infinity.^[1] Numerically, it is

\alpha =1.187452351126501\ldots

.^[2]

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

\lim _{n\to \infty }x_{n}{\frac {\log n}{n}}=1,

where "log" denotes the natural logarithm.^[1]

The same methods used in the proof of the uniqueness of the Foias constant may also be applied to other similar recursive sequences.^[3]

Notes and references

^ ^a ^b 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.
^ Sloane, N. J. A. (ed.). "Sequence A085848 (Decimal expansion of Foias's Constant)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Anghel, Nicolae (2018), "Foias numbers" (PDF), An. Ştiinţ. Univ. "Ovidius" Constanţa Ser. Mat., 26 (3): 21–28, doi:10.2478/auom-2018-0030, S2CID 195842026