The convergence and partial convergence of alternating series

Author:
J. R. Philip

Journal:
Math. Comp. **35** (1980), 907-916

MSC:
Primary 40A05; Secondary 65B10

MathSciNet review:
572864

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Abstract: The alternating series is , with *f* a single-signed monotonic function of the real variable *x*. The are , their sign fixed by repetition of the 'template' [*j*] of finite length 2*p*. [*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 2*pM* terms exists. For [*j*] 'pure', the improved Euler-Maclaurin expansion (IEM) gives the compact representation (A)

*D*th '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 Euler-Maclaurin 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:
https://doi.org/10.1090/S0025-5718-1980-0572864-8

Keywords:
Alternating series,
convergence,
practical summation

Article copyright:
© Copyright 1980
American Mathematical Society