PREPRINT: Riemann's Rearrangement Theorem in the Light of Hyperreal Numbers

Abstract

Riemann's rearrangement theorem has long been used to demonstrate that conditionally convergent series do not establish a single, coherent value. The theorem states that by simply regrouping the members of a conditionally divergent series, that a person can make that series converge to essentially any real number, or to diverge. Here, we will show that the problems usually associated with the rearrangement theorem disappear when using hyperreal numbers.