An Alternating Product Representation for Real Numbers

Abstract

In 1770 Lambert introduced two positive series expansions for the real numbers in terms of rationals. These were subsequently rediscovered by Sylvester (1880) and Engel (1913) after whom they are respectively named. A further positive series expansion for the real numbers was discovered by Luroth (1883). Also of particular interest to us is the product expansion of Cantor (1869). More recently, Oppenheim [3] defined a general algorithm for expressing real numbers in terms of a positive series of rational numbers. All of the previously mentioned expansions were shown to be special cases of the Oppenheim algorithm.