Foias constant

In mathematical analysis, the Foias constant is a real number named after Ciprian Foias. It is defined in the following way: for every real number x1 > 0, there is a sequence defined by the recurrence relation

x n + 1 = ( 1 + 1 x n ) n {\displaystyle x_{n+1}=\left(1+{\frac {1}{x_{n}}}\right)^{n}}

for n = 1, 2, 3, .... The Foias constant is the unique choice α such that if x1 = α then the sequence diverges to infinity. For all other values of x1, the sequence is divergent as well, but it has two accumulation points: 1 and infinity. Numerically, it is

α = 1.187452351126501 … {\displaystyle \alpha =1.187452351126501\ldots } . The constant can be computed by solving backwards:

x n = 1 x n + 1 n − 1 {\displaystyle x_{n}={\frac {1}{{\sqrt[{n}]{x_{n+1}}}-1}}}

This recursion converges for any starting value x n ≠ 0 {\displaystyle x_{n}\neq 0} . When x1 = α then the growth rate of the sequence (xn) is given by the limit

lim n → ∞ x n log ⁡ n n = 1 , {\displaystyle \lim _{n\to \infty }x_{n}{\frac {\log n}{n}}=1,}

where "log" denotes the natural logarithm. The same methods used in the proof of the uniqueness of the Foias constant may also be applied to other similar recursive sequences.

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