title: Foias constant
text: # 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 $x_1$ > 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 α such that if $x_1$ = α then the sequence diverges to infinity. For all other values of $x_1,$ 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 $x_1$ = α then the growth rate of the sequence $(x_n)$ 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.
  
  