Eikonal Blog

2010.02.09

A Knichin’s theorem on limit of geometric means for continued fraction form of (the most) real numbers

Filed under: mathematics — Tags: , — sandokan65 @ 15:57

Theorem: (Aleksandr Yakovlevich Khinchin; 1894-1959)
Let $\{a_n\in{\Bbb R}|n\in{\Bbb N}\}$ be a series of numbers. Define the impromptu geometric means $GM_n\{a\} :\equiv \left(a_1\cdot a_2 \cdots a_n\right)^{\frac1n}$. Then

$\lim_{n\rightarrow \infty} GM_n\{a\} = \prod_{k=1}^\infty \left(1+ \frac1{k(k+2)} \right)^{\frac{\ln(k)}{\ln(2)}} = 2.685,452,001,\cdots$

for almost all series $\{a\}$.

Since $\forall x\in{\Bbb R}$ one can write $x = [a_0; a_1, a_2, \cdots]$ (expansion of $x$ in the continued-fraction form), the above theorem is valid for almost all real $x$, except for:

• algebraic numbers with definite patterns for denominators,
• some transcendental numbers (e.g. $e$).

Source:

• Calvin C. Clawson “Mathematical Sorcery” (Revealing the secrets of numbers); ISBN 0-7382-0496-X; Perseus Publishing

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