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


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.

Share and Cite:

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.


[1] D. Bressoud, “A Radical Approach to Real Analysis,” Mathematical Association of America, Washington DC, 1994.
[2] L. Zhmud, “Pythagoras as a Mathematician,” Historia Mathematica, Vol. 16, No. 3, 1989, pp. 249-268. doi:10.1016/0315-0860(89)90020-7
[3] A. Lempner, “A Curious Convergent Series,” The American Mathematical Monthly, Vol. 21, No. 2, 1914, pp. 48-50. doi:10.2307/2972074
[4] M. Hoffman, “The Algebra of Multiple Harmonic Series,” Journal of Algebra, Vol. 194, No. 2, 1997, pp. 477-495. doi:10.1006/jabr.1997.7127
[5] M. E. Hoffman and C. Moen, “Sums of Triple Harmonic Series,” Journal of Number Theory, Vol. 60, No. 2, 1996, pp. 329-331. doi:10.1006/jnth.1996.0127
[6] G. Kawashima, “A Generalization of the Duality for Multiple Harmonic Sums,” Journal of Number Theory, Vol. 130, No. 2, 2010, pp. 347-359. doi:10.1016/j.jnt.2009.03.002
[7] H. Tsumura, “Multiple Harmonic Series Related to Multiple Euler Numbers,” Journal of Number Theory, Vol. 106, No. 1, 2004, pp. 155-168. doi:10.1016/j.jnt.2003.12.004
[8] D. Bradley, “Duality for Finite Multiple Harmonic q-Series,” Discrete Mathematics, Vol. 300, No. 1-3, 2005, pp. 44-56. doi:10.1016/j.disc.2005.06.008

Copyright © 2022 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.