Rational Numbers with Predictable Engel Product Expansions

Abstract

In a recent paper [4], Knopfmacher and Knopfmacher showed that every positive real number A < 1 has an expansion of the form

$$ A = \prod\limits_{{n = 1}}^{\infty } {\left( {1 - \frac{1}{{{a_1}{a_2} \cdots {a_n}}}} \right),} $$

(1)

where a _n is a positive integer for n ≥ 1, a ₁ ≥ 2, a _{n + 1} ≥ a _n − 1 for n ≥ 1 and a _n ≥ 2 infinitely often.