Foias constant

In mathematical analysis, the Foias constant is a real number named after Ciprian Foias.

Evolution of the sequence ${\displaystyle x_{n+1}=(1+1/x_{n})^{n}}$ for several values of ${\displaystyle x_{1}}$, around the Foias constant ${\displaystyle \alpha }$. Evolution for ${\displaystyle x_{1}=\alpha }$ is in green. Other initial values lead to two accumulation points, 1 and ${\displaystyle \infty }$. A logarithmic scale is used.

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]

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

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

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]