Foias constant

In mathematical analysis, the 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 α such that if x₁ = α then the sequence diverges to ∞.^[1] Numerically, it is

\alpha =1.187452351126501\ldots \,

No closed form is known.

When x₁ = α then we have the limit:

\lim _{{n\to \infty }}x_{n}{\frac {\log n}n}=1,

where "log" denotes the usual natural logarithm.

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

\lim _{{n\to \infty }}{\frac {x_{n}}{\pi (n)}}=1,

Notes and references

↑ 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.
↑ 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.

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