Almost Alternating Sums

Details Almost Alternating Sums Almost Alternating Sums

2003-08-09

arxiv.org/abs/math/0308087v2

Amer. Math. Monthly 113 (2006), no. 8, 673--688.

Mathematics

Number Theory

17 pages, 3 figures (revision is typographical)

Scientific paper

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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