Definition:Foiaș Constant/Second

Definition
Let $x_1 \in \R_{>0}$ be a (strictly) positive real number.

Let:
 * $x_{n + 1} = \left({1 + \dfrac 1 {x_n} }\right)^n$

for $n = 1, 2, 3, \ldots$

The second Foiaș constant is defined as the unique real number $\alpha$ such that if $x_1 = \alpha$ then the sequence $\left\langle{x_{n + 1} }\right\rangle$ diverges to infinity.

No closed-form expression is known.

Also known as
The second Foiaș constant is also known as just the Foiaș constant.

Some sources refer to it as Foiaș' constant.

Many sources omit the diacritic: Foias.

When $x_1 = \alpha$ then we have the limit:


 * $\displaystyle \lim_{n \mathop \to \infty} x_n \frac{\ln n} n = 1$