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 an where the value of each an is determined by the flip of a coin: heads on the nth toss will mean that an = 1 and tails that an = −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

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