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:

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

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.