Evolution of the sequence x n + 1 = ( 1 + 1 / x n ) n {\displaystyle x_{n+1}=(1+1/x_{n})^{n}} for several values of x 1 {\displaystyle x_{1}}, around the Foias constant α {\displaystyle \alpha }. Evolution for x 1 = α {\displaystyle x_{1}=\alpha } is in green. Other initial values lead to two accumulation points, 1 and ∞ {\displaystyle \infty }. A logarithmic scale is used.

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 }.

No closed form for the constant is known.

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.

See also

Notes and references

  • S. R. Finch (2003). . Cambridge University Press. p. . ISBN 0-521-818-052. Foias constant.