Cahen's constant

In mathematics, Cahen's constant is defined as an infinite series of unit fractions, with alternating signs, derived from Sylvester's sequence:

${\displaystyle C=\sum {\frac {(-1)^{i}}{s_{i}-1}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{6}}-{\frac {1}{42}}+{\frac {1}{1806}}-\cdots \approx 0.64341054629.}$

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:

${\displaystyle C=\sum {\frac {1}{s_{2i}}}={\frac {1}{2}}+{\frac {1}{7}}+{\frac {1}{1807}}+{\frac {1}{10650056950807}}+\cdots }$

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 & 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

0, 1, 1, 1, 2, 3, 14, 129, 25298, 420984147, ... (sequence A006279 in the OEIS)

defined by the recurrence relation

${\displaystyle a_{0}=0,~a_{1}=1,~a_{n+2}=a_{n}\left(1+a_{n}a_{n+1}\right)~\forall ~n\in \mathbb {Z} _{\geqslant 2}}$

then Cahen's constant has a canonical continued fraction of:

${\displaystyle \left[a_{0}^{2};a_{1}^{2},a_{2}^{2},a_{3}^{2},a_{4}^{2},\ldots \right]}$

(Davison & Shallit 1991).