# 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

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

defined by the recurrence relation

${\displaystyle q_{n+2}=q_{n}^{2}q_{n+1}+q_{n}}$

then the continued fraction expansion of Cahen's constant is

${\displaystyle [0,1,q_{0}^{2},q_{1}^{2},q_{2}^{2},\ldots ]}$