Definition:Foiaș Constant/Second
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Definition
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.
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.
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When $x_1 = \alpha$ then we have the limit:
- $\ds \lim_{n \mathop \to \infty} x_n \frac {\ln n} n = 1$
Source of Name
This entry was named for Ciprian Ilie Foiaș.
Sources
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