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| In [[mathematics]], '''Cahen's constant''' is defined as an [[Series (mathematics)|infinite series]] of [[unit fraction]]s, with alternating signs, derived from [[Sylvester's sequence]]:
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| :<math>C = \sum\frac{(-1)^i}{s_i-1}=\frac11 - \frac12 + \frac16 - \frac1{42} + \frac1{1806} - \cdots\approx 0.64341054629.</math>
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| 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 algorithm for Egyptian fractions|greedy Egyptian expansion]]:
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| :<math>C = \sum\frac{1}{s_{2i}}=\frac12+\frac17+\frac1{1807}+\frac1{10650056950807}+\cdots</math>
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| This constant is named after Eugène Cahen (also known for the [[Cahen-Mellin integral]]), who first formulated and investigated its series {{harv|Cahen|1891}}.
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| Cahen's constant is known to be [[Transcendental number|transcendental]] {{harv|Davison|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
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| :1, 1, 2, 3, 14, 129, 25298, 420984147, ... {{OEIS|id=A006279}}
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| defined by the [[recurrence relation]]
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| :<math>q_{n+2} = q_n^2 q_{n+1} + q_n</math>
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| then the continued fraction expansion of Cahen's constant is
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| :<math>[0,1,q_0^2,q_1^2,q_2^2,\ldots]</math> | |
| {{harv|Davison|Shallit|1991}}.
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| == References ==
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| *{{citation
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| | last = Cahen | first = Eugène
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| | title = Note sur un développement des quantités numériques, qui présente quelque analogie avec celui en fractions continues
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| | journal = Nouvelles Annales de Mathématiques
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| | volume = 10
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| | year = 1891
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| | pages = 508–514}}
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| *{{citation
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| | last1 = Davison | first1 = J. Les | author2-link = Jeffrey Shallit | last2 = Shallit | first2 = Jeffrey O.
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| | title = Continued fractions for some alternating series
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| | journal = Monatshefte für Mathematik
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| | volume = 111
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| | year = 1991
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| | pages = 119–126
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| | doi = 10.1007/BF01332350
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| | issue = 2}}
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| == External links ==
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| *{{mathworld | title = Cahen's Constant | urlname = CahensConstant}}
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| *{{citation | title = The Cahen constant to 4000 digits | url = http://pi.lacim.uqam.ca/piDATA/cahen.txt | work = Plouffe's Inverter | publisher = [[Université du Québec à Montréal]] | accessdate= 2011-03-19}}
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| [[Category:Transcendental numbers]]
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| [[Category:Mathematical constants]]
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