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 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.


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— base constant
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