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Liouville's Constant

L\equiv \sum_{n=1}^\infty 10^{-n!}=0.110001000000000000000001\ldots

(Sloane's A012245). Liouville's constant is a decimal fraction with a 1 in each decimal place corresponding to a Factorial $n!$, and Zeros everywhere else. This number was among the first to be proven to be Transcendental. It nearly satisfies


but with $x=L$, this equation gives $-0.00000 00059\ldots$.

See also Liouville Number


Conway, J. H. and Guy, R. K. ``Liouville's Number.'' In The Book of Numbers. New York: Springer-Verlag, pp. 239-241, 1996.

Courant, R. and Robbins, H. ``Liouville's Theorem and the Construction of Transcendental Numbers.'' §2.6.2 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 104-107, 1996.

Liouville, J. ``Sur des classes très étendues de quantités dont la valeur n'est ni algébrique, ni même reductible à des irrationelles algébriques.'' C. R. Acad. Sci. Paris 18, 883-885 and 993-995, 1844.

Liouville, J. ``Sur des classes très-étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationelles algébriques.'' J. Math. pures appl. 15, 133-142, 1850.

Sloane, N. J. A. Sequence A012245 in ``The On-Line Version of the Encyclopedia of Integer Sequences.''

© 1996-9 Eric W. Weisstein