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

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Author(s): Christopher M. Davis | David G. Taylor

Journal: Advances in Pure Mathematics
ISSN 2160-0368

Volume: 03;
Issue: 03;
Start page: 309;
Date: 2013;
Original page

Keywords: Harmonic Series | Biconvergence | Non-Integer Series

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.
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