# Definition:Foiaș Constant/Second

< Definition:Foiaș Constant(Redirected from Definition:Second Foiaș Constant)

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## 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.