Theorem: (Aleksandr Yakovlevich Khinchin; 1894-1959)
Let be a series of numbers. Define the impromptu geometric means . Then
for almost all series .
Since one can write (expansion of in the continued-fraction form), the above theorem is valid for almost all real , except for:
- algebraic numbers with definite patterns for denominators,
- some transcendental numbers (e.g. ).
- Calvin C. Clawson “Mathematical Sorcery” (Revealing the secrets of numbers); ISBN 0-7382-0496-X; Perseus Publishing