The Divergence of Balanced Harmonic-like Series

Authors: Lutzer, Carl V.; Marengo, James E.

Source: The College Mathematics Journal, Volume 37, Number 5, November 2006 , pp. 364-369(6)

Publisher: Mathematical Association of America

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Abstract:

Consider the series Σ^∞_n=1 a_n where the value of each a_n is determined by the flip of a coin: heads on the n^th toss will mean that a_n = 1 and tails that a_n = −1. Assuming that the coin is "fair," what is the probability that this harmonic-like series converges? After a moment's thought, many people answer that the probability of convergence is 1. This is correct (though the proof is nontrivial), but it doesn't preclude the existence of a divergent example. Indeed, Feist and Naimi provided just such an example in 2004. In this paper, we construct an uncountably infinite family of examples as a companion result.

Document Type: Research article

Publication date: 2006-11-01

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The College Mathematics Journal is designed to enhance classroom learning and stimulate thinking regarding undergraduate mathematics. CMJ publishes articles, short Classroom Capsules, problems, solutions, media reviews and other pieces. All are aimed at the college mathematics curriculum with emphasis on topics taught in the first two years.
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