Foias constant

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.^[1] Numerically, it is

\alpha =1.187452351126501\ldots

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

\lim _{n\to \infty }{\frac {x_{n}}{\pi (n)}}=1,

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.

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