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

${\displaystyle x_{n+1}=\left(1+{\frac {1}{x_{n}}}\right)^{n}}$

for n = 1, 2, 3, .... The Foias constant is the unique choice α such that if x₁ = α then the sequence diverges to infinity. 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

${\displaystyle \alpha =1.187452351126501\ldots }$.^[2]

The constant can be computed by solving backwards: ${\displaystyle x_{n}={\frac {1}{{\sqrt[{n}]{x_{n+1}}}-1}}}$

This recursion converges for any starting value ${\displaystyle x_{n}\neq 0}$^[2].

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

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

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]