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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
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 $ {S_j}$, or some suitable modification $ {S'_j}$, closely related to $ {S_j}$, as a polynomial in 1/j. Either $ {S_\infty }$ or $ {S_n}$ was found by m-point Lagrangian extrapolation from $ {{S}_{{{j}_{0}}}}$, $ {{S}_{{{j}_{0}}-1}}$, $ \cdot \cdot \cdot $, $ {{S}_{{{j}_{0}}-m+1}}$ to 1/j = 0 or 1/j = 1/n respectively. This present paper furnishes more accurate m-point formulas for sums (or sequences) $ {S_j}$ which behave as even functions of 1/j. Tables of Lagrangian extrapolation coefficients in the variable $ 1/{j^2}$ are given for: complete summation, m = 2(1)7, $ {j_0}$ = 10, exactly and 20D, m = 11, $ {j_0}$ = 20, 30D; partial summation, m = 7, $ {j_0}$ = 10, n = 11(1)25(5)100, 200, 500, 1000, exactly. Applications are made to calculating $ \pi $ or the semi-perimeters of many-sided regular polygons, Euler's constant,

$\displaystyle 1 + \sum\limits_{r = 1}^j {\left\{ {\tfrac{{ - 1}}{{{{(4r - 1)}^2... ... \tfrac{1}{{{3^2}}} + \tfrac{1}{{{5^2}}} - \cdots } \,{\text{for}}\,j = \infty $

(Catalan's constant), calculation of a definite integral as the limit of a suitably chosen sequence, determining later zeros of $ {J_v}(x)$ from earlier zeros for suitable v, etc. A useful device in many cases involving sums of odd functions, is to replace $ {S_j}$ by a trapezoidal-type $ {S'_j}$ which is seen, from the Euler-Maclaurin formula, to be formally a series in $ 1/{j^2}$. 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.

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

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DOI: https://doi.org/10.1090/S0025-5718-1961-0121972-2
Article copyright: © Copyright 1961 American Mathematical Society