CahensConstant 2006-11-12 15:57:42 2006-11-12 16:45:29 Definition Sylvester's sequence % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Whereas a simple addition of unit fractions with the terms of Sylvester's sequence as denominators gives as a result the integer 1, an alternating sum $$\sum_{i = 0}^\infty \frac{(-1)^i}{a_i - 1}$$ (where $a_i$ is the $i$th term of Sylvester's sequence) gives the transcendental number known as \emph{Cahen's constant} (after Eug\`ene Cahen) with an approximate decimal value of 0.643410546288338026182254307757564763286587860268239505987 (see A118227 in Sloane's OEIS). Alternatively, we can express Cahen's constant as $$\sum_{j = 0}^\infty \frac{1}{a_{2j}}.$$ The recurrence relation $b_{n + 2} = {b_n}^2b_{n + 1} + b_n$ gives us the terms for the continued fraction representation of this constant: $$1 + \frac{1}{{b_0}^2 + \frac{1}{{b_1}^2 + \frac{1}{{b_3}^2 + \, \cdots}}}$$