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

DOI: 10.4236/apm.2013.33044   PDF   HTML     3,468 Downloads   6,047 Views  


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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C. Davis and D. Taylor, "One Step Forward, Two Steps Back: Biconvergence of Washed Harmonic Series," Advances in Pure Mathematics, Vol. 3 No. 3, 2013, pp. 309-316. doi: 10.4236/apm.2013.33044.

Conflicts of Interest

The authors declare no conflicts of interest.


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