Rearrangements of a Conditionally Convergent Series Summing to Logarithms of Natural Numbers

Abstract

It is a counterintuitive idea to calculus students that conditionally convergent series may be rearranged to converge to different sums. Some nice examples can be helpful and fascinating to students. This note describes a family of such rearrangements suitable for calculus and undergraduate analysis students. There is one series for each k ∈ N , with k = 1 the original series. The cases of k = 1 and k = 2 are separately presented, as they are particularly easy to show to a calculus class. The general case uses material from a first calculus class but is more involved. Various versions of the series are known and, for k > 1 , wonderful. For each series (labeled k ∈ N ) the positive terms and the negative terms form two harmonic series. The order of the two series are preserved so it is easy to see they are rearrangements.