Improved formulas for complete and partial summation of certain series.

Authors:
Herbert E. Salzer and Genevieve M. Kimbro

Journal:
Math. Comp. **15** (1961), 23-39

MSC:
Primary 65.00

DOI:
https://doi.org/10.1090/S0025-5718-1961-0121972-2

MathSciNet review:
0121972

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Abstract: In two previous articles one of the authors gave formulas, with numerous examples, for summing a series either to infinity (complete) or up to a certain number *n* of terms (partial) by considering the sum of the first *j* terms , or some suitable modification , closely related to , as a polynomial in 1/*j*. Either or was found by *m*-point Lagrangian extrapolation from , , , to 1/*j* = 0 or 1/*j* = 1/*n* respectively. This present paper furnishes more accurate *m*-point formulas for sums (or sequences) which behave as even functions of 1/*j*. Tables of Lagrangian extrapolation coefficients in the variable are given for: complete summation, *m* = 2(1)7, = 10, exactly and 20D, *m* = 11, = 20, 30D; partial summation, *m* = 7, = 10, *n* = 11(1)25(5)100, 200, 500, 1000, exactly. Applications are made to calculating or the semi-perimeters of many-sided regular polygons, Euler's constant,

*v*, etc. A useful device in many cases involving sums of odd functions, is to replace by a trapezoidal-type which is seen, from the Euler-Maclaurin formula, to be formally a series in . In almost every example, comparison with the earlier (1/

*j*)-extrapolation showed these present formulas, for 7 points, to improve results by anywhere from around 4 to 9 places.

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DOI:
https://doi.org/10.1090/S0025-5718-1961-0121972-2

Article copyright:
© Copyright 1961
American Mathematical Society