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In [[mathematical analysis]], the '''Foias constant''', is a number named after [[Ciprian Foias]].

If ''x''1 > 0 and

: <math> x_{n+1} = \left( 1 + \frac{1}{x_n} \right)^n\text{ for }n=1,2,3,\ldots,</math>

then the Foias constant is the unique [[real number]] ''α'' such that if ''x''1 = ''α'' then the sequence [[divergent sequence|diverges]] to ∞.<ref>Ewing, J. and Foias, C. "An Interesting Serendipitous Real Number." In ''Finite versus Infinite: Contributions to an Eternal Dilemma'' (Ed. C. Caluse and G. Păun). London: Springer-Verlag, pp. 119–126, 2000.</ref> Numerically, it is

: <math> \alpha = 1.187452351126501\ldots\, </math> {{OEIS2C|id=A085848}}.

No [[closed-form expression|closed form]] is known.

When ''x''1 = ''α'' then we have the [[Limit of a sequence|limit]]:

: <math> \lim_{n\to\infty} x_n \frac{\log n}n = 1, </math>

where "log" denotes the usual [[natural logarithm]].

A fortuitous observation between the [[prime number theorem]] and this constant goes as follows,

: <math> \lim_{n\to\infty} \frac{x_n}{\pi(n)} = 1, </math>

where {{pi}} is the [[prime-counting function]].<ref>Ewing, J. and Foias, C. "An Interesting Serendipitous Real Number." In ''Finite versus Infinite: Contributions to an Eternal Dilemma'' (Ed. C. Caluse and G. Păun). London: Springer-Verlag, pp. 119–126, 2000.</ref>

==See also==

*[[Grossman's constant]]
*[[Mathematical constant]]

== Notes and references ==

{{reflist}}
* {{MathWorld |title=Foias Constant |id=FoiasConstant}}

* {{cite book|title=Mathematical Constants|page=430|publisher=Cambridge University Press|year=2003|isbn=0-521-818-052|author=S. R. Finch|url=http://books.google.co.uk/books?id=Pl5I2ZSI6uAC&pg=PA595&lpg=PA595&dq=Foias+constant&source=bl&ots=K0oI60ku_c&sig=NkJa9PImpPWG0283ymbqjyG2f6Y&hl=en&sa=X&ei=fBoJUuDnDcqG0AWYx4GADg&ved=0CFkQ6AEwBw#v=onepage&q=Foias%20constant&f=false}}

* {{SloanesRef |sequencenumber=A085848 |name=Decimal expansion of Foias's Constant}}

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[[Category:Mathematical analysis]]
[[Category:Mathematical constants]]

Chebyshev integral - Revision history

en>David Eppstein: {{mathanalysis-stub}}