TITLE:
One Step Forward, Two Steps Back: Biconvergence of Washed Harmonic Series
AUTHORS:
Christopher M. Davis, David G. Taylor
KEYWORDS:
Harmonic Series; Biconvergence; Non-Integer Series
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.3 No.3,
May
8,
2013
ABSTRACT:
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.