Definition:Foiaș Constant

The second Foias constant, is a number named after Ciprian Foias.

If x1 > 0 and


 * $$ x_{n+1} = \left( 1 + \frac{1}{x_n} \right)^n\text{ for }n=1,2,3,\ldots,$$

then the Foias constant is the unique real number &alpha; such that if x1 = &alpha; then the sequence diverges to &infin;.


 * $$ \alpha = 1.187452351126501\ldots\, $$.

No closed-form expression is known.

When x1 = &alpha; then we have the limit:


 * $$ \lim_{n\to\infty} x_n \frac{\ln n}n = 1, $$.

The first Foias constant, is also named after Ciprian Foias. If : $$ x_{n+1} = \left( 1 + \frac{1}{x_n} \right)^{x_n}\text{ for }n=1,2,3,\ldots,$$

then the first Foias constant is the limit    : $$ x_{\infty} = 2.293166287411861031508028291250805864372257290327121248537 \ldots\ $$