Cahen's constant
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In mathematics, Cahen's constant is defined as an infinite series of unit fractions, with alternating signs, derived from Sylvester's sequence:
By considering these fractions in pairs, we can also view Cahen's constant as a series of positive unit fractions formed from the terms in even positions of Sylvester's sequence; this series for Cahen's constant forms its greedy Egyptian expansion:
This constant is named after Eugène Cahen (also known for the Cahen-Mellin integral), who first formulated and investigated its series (Cahen 1891).
Cahen's constant is known to be transcendental (Davison and Shallit 1991). It is notable as being one of a small number of naturally occurring transcendental numbers for which we know the complete continued fraction expansion: if we form the sequence
defined by the recurrence
then the continued fraction expansion of Cahen's constant is
![[0,1,q_0^2,q_1^2,q_2^2,\ldots]](../../../math/9/0/a/90ac86fcb56ede771836464c99ff8a7f.png)
(Davison and Shallit 1991).
[edit] References
- Cahen, Eugène (1891). "Note sur un développement des quantités numériques, qui présente quelque analogie avec celui en fractions continues". Nouvelles Annales de Mathématiques 10: 508–514.
- Davison, J. Les; Shallit, Jeffrey O. (1991). "Continued fractions for some alternating series". Monatshefte für Mathematik 111: 119–126. DOI:10.1007/BF01332350.
[edit] External links
- Weisstein, Eric W. Cahen's Constant. MathWorld–A Wolfram Web Resource.
