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

Affiliation(s)

Department of Mathematics, George Mason University, Fairfax, USA.

MCSP Department, Roanoke College, Salem, USA.

Department of Mathematics, George Mason University, Fairfax, USA.

MCSP Department, Roanoke College, Salem, USA.

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.

Cite this paper

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.

C. Davis and D. Taylor, "One Step Forward, Two Steps Back: Biconvergence of Washed Harmonic Series,"

References

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[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

[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