Almost Alternating Sums
Number Theory
2007-05-23 v2
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}, ....
Cite
@article{arxiv.math/0308087,
title = {Almost Alternating Sums},
author = {Kevin O'Bryant and Bruce Reznick and Monika Serbinowska},
journal= {arXiv preprint arXiv:math/0308087},
year = {2007}
}
Comments
17 pages, 3 figures (revision is typographical)