Almost Alternating Sums
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 Σn=1N (-1)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=2, 1+5, 2+10, ....
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