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Almost Alternating Sums

Kevin O'Bryant·Bruce Reznick·Monika Serbinowska
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Abstract

Writing for a general mathematical audience, we provide elementary upper and lower bounds on the growth (as a function of N) of the sum \sum_{n=1}^N (-1)^{\floor{n x}} for various fixed x. For example, if x is a quadratic irrational, then the sum is O(log N), and if x is 2/(e-1), then the sum is O(log N / log log N). We compute the optimal big-Oh constant for x=\sqrt{2}, 1+\sqrt{5}, 2+\sqrt{10}, ....

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Kevin O'Bryant, Bruce Reznick, Monika Serbinowska. 2005-08-24. Almost Alternating Sums. https://arxiv.org/abs/math/0308087

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