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Cahen's constant

In mathematics, Cahen's constant is defined as the value of an infinite series of unit fractions with alternating signs:
${\\displaystyle C=\\sum _{i=0}^{\\infty }{\ rac {(-1)^{i}}{s_{i}-1}}={\ rac {1}{1}}-{\ rac {1}{2}}+{\ rac {1}{6}}-{\ rac {1}{42}}+{\ rac {1}{1806}}-\\cdots \\approx 0.643410546288...}$ (sequence A118227 in the OEIS)
Here ${\\displaystyle (s_{i})_{i\\geq 0}}$ denotes Sylvester's sequence, which is defined recursively by
${\\displaystyle {\egin{array}{l}s_{0}~~~=2;\\\\s_{i+1}=1+\\prod _{j=0}^{i}s_{j}{\ ext{ for }}i\\geq 0.\\end{array}}}$
Combining these fractions in pairs leads to an alternative expansion of 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:
${\\displaystyle C=\\sum {\ rac {1}{s_{2i}}}={\ rac {1}{2}}+{\ rac {1}{7}}+{\ rac {1}{1807}}+{\ rac {1}{10650056950807}}+\\cdots }$
This constant is named after Eugène Cahen (also known for the Cahen–Mellin integral), who was the first to introduce it and prove its irrationality.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Cahen%27s_constant)

constante de Cahen

French

URI

http://data.loterre.fr/ark:/67375/PSR-HXC2CCN6-1

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