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Convergence acceleration of alternating series. (English) Zbl 0972.11115
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.

##### MSC:
 11Y35 Analytic computations 65B10 Numerical summation of series
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