# 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.

### Decimal Expansion

The decimal expansion of the second Foiaș Constant starts:

$\alpha = 1 \cdotp 18745 \, 23511 \, 26501 \ldots$

## 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.

## Also see

• Results about the Foiaș constants can be found here.

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

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

## Source of Name

This entry was named for Ciprian Ilie Foiaș.