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The Divergence of Balanced Harmonic-like Series

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Abstract:

Consider the series Σn=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

Publication date: November 1, 2006

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