# Definition:Foiaș Constant

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This page needs proofreading.In particular: Check that the actual attributed numbers "first" and "second" for this pair of constants has been defined correctly. I believe the way it has been defined here is wrong. The author of the original page is requested to learn house style before embarking on substantial contributions -- putting it into a readable and usable format is time-consuming and tedious.If you believe all issues are dealt with, please remove `{{Proofread}}` from the code.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Proofread}}` from the code. |

## 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:

- $x_{n + 1} = \paren {1 + \dfrac 1 {x_n} }^n$

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 $\sequence {x_{n + 1} }$ 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.