Cohen, Henri; Rodriguez Villegas, Fernando; Zagier, Don Convergence acceleration of alternating series. (English) Zbl 0972.11115 Exp. Math. 9, No. 1, 3-12 (2000). This paper treats several algorithms to calculate the sum of an alternating series \(\sum_{k=0}^{\infty} (-1)^ka_k\) from the first \(n\) entries. The \(a_k\) have to be either the moments of a positive measure on \([0,1]\) or those of a weight \(w(x)\) on \([0,1]\) that extends analytically into the complex domain in a certain manner. In the first case the relative accuracy is of the order \(5.828^{-n}\); the algorithm uses \(O(1)\) storage and has a running time of \(O(1)\) for each \(a_k\) used. Chebyshev polynomials play a role in the construction of the algorithm. The second algorithm requires storage \(O(n)\) and running time \(O(n^2)\); the relative accuracy is of the order \(7.89^{-n}\) for a large class of series and \(17.93^{-n}\) for a smaller class. The conclusion is, that the choice of algorithm depends on the ease with which the terms of the series can be computed. A very interesting and nicely written paper. Reviewer: Marcel G.de Bruin (Delft) Cited in 23 Documents MSC: 11Y35 Analytic computations 65B10 Numerical summation of series Keywords:convergence acceleration; alternating sum; Chebyshev polynomial; algorithms × Cite Format Result Cite Review PDF Full Text: DOI EuDML Link Online Encyclopedia of Integer Sequences: Decimal expansion of Sum_{k>=1} (-1)^k*log(k)/k. Number of 9’s immediately following the decimal point in the expansion of (3+sqrt(8))^n. Number of correct digits of the MRB constant derived from the sequence of partial sums up to m=10^n terms as defined by S[n]= Sum[(-1)^k*(k^(1/k)-1),{k,m}]. Decimal expansion of the ”value” -Sum_{n>=1} (-1)^n / n^(1/n). Decimal expansion of log(sqrt(Pi/2)). Decimal expansion of Sum_{k>=1} (-1)^k*log(k)/sqrt(k). Decimal expansion of Product_{k>=1} Gamma(2k/(2k-1)) / Gamma(1+1/(2k)). References: [1] Borwein, J. and Crandall, R. [Borwein and Crandall 2000], In preparation [2] Brezinski C., Padé-type approximation and general orthogonal polynomials. (1980) · Zbl 0418.41012 [3] Eiermann M., J. Comput. Appl. Math. 10 (2) pp 219– (1984) · Zbl 0538.65011 · doi:10.1016/0377-0427(84)90059-1 [4] Gustafson S.-A., Computing 21 (1) pp 53– (1978) · Zbl 0403.65002 · doi:10.1007/BF02252194 [5] DOI: 10.1007/BF01175684 · JFM 49.0193.01 · doi:10.1007/BF01175684 [6] Hurwitz A., Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen (1929) [7] Press W. H., Numerical recipes in C,, 2. ed. (1988) · Zbl 0661.65001 [8] Wimp J., Sequence transformations and their applications (1981) · Zbl 0566.47018 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.