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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 $ \Sigma _{n = 1}^\infty {j_n}f(n) = [j]f$, with f a single-signed monotonic function of the real variable x. The $ {j_n}$ are $ \pm 1$, 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 $ [j]f$ is 'partially convergent' if and only if $ {\lim _{x \to \infty }}{f^{(D - 1)}}(x)$ is bounded. A series is partially convergent when the limit as $ M \to \infty $ of the sum of 2pM terms exists. For [j] 'pure', the improved Euler-Maclaurin expansion (IEM) gives the compact representation (A)

$\displaystyle {S^{(p)}} \sim - \frac{{\mu (D)}}{{2p}}\sum\limits_{r = 0}^\infty... ... {f^{(2r + D - 1)}}({\theta _r}),\quad 1 - p \leqslant {\theta _r} \leqslant p.$

$ {S^{(p)}}$ is the sum, $ \mu (D)$ is the Dth 'template moment', and the $ {B_{2r}}$ 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 $ {\theta _0} = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} $ gives a simple but effective estimate of sums. The paper also gives a comparison with Euler's transformation in the case $ p = 1$ and discusses sums to N terms with $ N/2p$ nonintegral and finite but large.

References [Enhancements On Off] (What's this?)

  • [1] K. KNOPP, Theory and Application of Infinite Series, 2nd ed., Blackie, London, 1951.
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Keywords: Alternating series, convergence, practical summation
Article copyright: © Copyright 1980 American Mathematical Society

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