Geometric Mean of Continued Simple Fraction Terms

A remarkable theorem of Khinchin states that for almost all real numbers , the simple continued fraction of ,, has the property that the geometric mean approaches a universal constant, the Khinchin constant ≈ 2.69, as tends to infinity. This constant is shown as a horizontal line in the graph. Interestingly, not a single naturally occuring number is known for which one can verify this property. (Rational numbers, quadratic surds, and powers of ⅇ are known exceptions.) This Demonstration explores the convergence behavior of the first 1000 continued fraction terms for simple multiples of powers of various transcendental numbers.