# Definition:Foiaș Constant

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## Definition

### First Foiaș Constant

Let:

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

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

The **first Foiaș constant** is the limit of $x_n$ as $n \to \infty$.

### Second Foiaș Constant

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.

## Also known as

Many sources omit the diacritic: **Foias**.

Some sources refer to the **second Foiaș constant** as **the Foiaș constant**.

## Source of Name

This entry was named for Ciprian Ilie Foiaș.