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
 * $$ x_{n+1} = \left( 1 + \frac{1}{x_n} \right)^n$$

for n = 1, 2, 3, .... The Foias constant is the unique choice &alpha; such that if x1 = &alpha; 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
 * $$ \alpha = 1.187452351126501\ldots$$.

No closed form for the constant is known.

When x1 = &alpha; then the growth rate of the sequence (xn) is given by the limit
 * $$ \lim_{n\to\infty} x_n \frac{\log n}n = 1, $$

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.