On the alternating series 1− 12 + 13 − 14 +···

It is shown that the best constants a and b such that inequalities 1 2n+a | ∞k =n+1(−1)k−1 1 k | < 1 2n+b hold for every n 1 are a = 1 1−log2 −2 ≈ 1.258891 and b = 1.  2003 Elsevier Science (USA). All rights reserved