Abstract
The best possible bounds for the sum of an alternating series with completely monotonic terms, when 2N terms have been computed, are determined. It is shown that their difference decreases exponentially withN. Various generalizations are indicated. The optimal application of Euler's transformation is also discussed. The error of that method also decreases exponentially, though the logarithmic decrement is only about 2/3 compared with the best possible error bounds.
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Dahlquist, G., Gustafson, SÅ. & Siklósi, K. Convergence acceleration from the point of view of linear programming. BIT 5, 1–16 (1965). https://doi.org/10.1007/BF01975719
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DOI: https://doi.org/10.1007/BF01975719