One Step Forward, Two Steps Back: Biconvergence of Washed Harmonic Series

• 作者: Christopher M. Davis David G. Taylor
• 来源期刊: 理论数学进展(英文)
• 学科: 数学
• 关键词: HARMONIC SERIES Biconvergence Non-Integer SERIES
• 年，卷（期）: 2013,（3）
• 页码范围: 309-316
• 页数: 8页
• 分类号: O1

摘要

We examine variations of the harmonic series by grouping terms into “washings” that alternate sign with the number of terms in a washing growing exponentially with respect to a fixed base. The bases x = 1 and x = ∞ correspond to the alternating harmonic series and the usual harmonic series;we first consider other positive integral bases and further we consider positive real number bases with a unique way to make sense of adding a non-integral number of terms together. In both cases, we prove a remarkable result regarding the difference between the upper and lower convergent values of the series, and give some analysis of this behavior.