The convergence and partial convergence of alternating series
Author:
J. R. Philip
Journal:
Math. Comp. 35 (1980), 907916
MSC:
Primary 40A05; Secondary 65B10
MathSciNet review:
572864
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Abstract: The alternating series is , with f a singlesigned monotonic function of the real variable x. The are , their sign fixed by repetition of the 'template' [j] of finite length 2p. [j] constitutes a difference scheme of 'differential order' D, which can be determined. The principal theorem is that is 'partially convergent' if and only if is bounded. A series is partially convergent when the limit as of the sum of 2pM terms exists. For [j] 'pure', the improved EulerMaclaurin expansion (IEM) gives the compact representation (A) is the sum, is the Dth 'template moment', and the are Bernoulli numbers. Efficient means for practical summation of these series follow also from IEM. In illustration, 10 alternating series with D ranging from 1 to 3 are summed using IEM. It is found that the leading term of (A) with gives a simple but effective estimate of sums. The paper also gives a comparison with Euler's transformation in the case and discusses sums to N terms with nonintegral and finite but large.
 [1]
K. KNOPP, Theory and Application of Infinite Series, 2nd ed., Blackie, London, 1951.
 [2]
T. J. ÍA. BROMWICH, An Introduction to the Theory of Infinite Series, 2nd ed., Macmillan, London, 1926.
 [3]
G. H. HARDY, Orders of Infinity, 2nd ed., Cambridge Univ. Press, Cambridge, 1924.
 [4]
J. R. PHILIP, "The symmetrical EulerMaclaurin summation formula," The Mathematical Scientist. (In press.)
 [5]
M. ABRAMOWITZ, "3. Elementary analytical methods," in Handbook of Mathematical Functions (M. Abramowitz & I. A. Stegun, Eds.), Nat. Bur. Standards Appl. Math. Series 55, U. S. Dept. of Commerce, Washington, D. C., 1964. MR 29 #4914.
 [1]
 K. KNOPP, Theory and Application of Infinite Series, 2nd ed., Blackie, London, 1951.
 [2]
 T. J. ÍA. BROMWICH, An Introduction to the Theory of Infinite Series, 2nd ed., Macmillan, London, 1926.
 [3]
 G. H. HARDY, Orders of Infinity, 2nd ed., Cambridge Univ. Press, Cambridge, 1924.
 [4]
 J. R. PHILIP, "The symmetrical EulerMaclaurin summation formula," The Mathematical Scientist. (In press.)
 [5]
 M. ABRAMOWITZ, "3. Elementary analytical methods," in Handbook of Mathematical Functions (M. Abramowitz & I. A. Stegun, Eds.), Nat. Bur. Standards Appl. Math. Series 55, U. S. Dept. of Commerce, Washington, D. C., 1964. MR 29 #4914.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198005728648
PII:
S 00255718(1980)05728648
Keywords:
Alternating series,
convergence,
practical summation
Article copyright:
© Copyright 1980
American Mathematical Society
