Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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, \[ 1 + \sum \limits _{r = 1}^j {\left \{ {\tfrac {{ - 1}}{{{{(4r - 1)}^2}}} + \tfrac {1}{{{{(4r + 1)}^2}}}} \right \} = 1 - \tfrac {1}{{{3^2}}} + \tfrac {1}{{{5^2}}} - \cdots } {\text {for}} j = \infty \] (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?)

  • Herbert E. Salzer, A simple method for summing certain slowly convergent series, J. Math. and Phys. 33 (1955), 356–359. MR 68315, DOI
  • Herbert E. Salzer, Formulas for the partial summation of series, Math. Tables Aids Comput. 10 (1956), 149–156. MR 81528, DOI
  • L. F. Richardson, “The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam,” Philos. Trans. Roy. Soc. London, Ser. A, v. 210, 1910, p. 307-357. L. F. Richardson & J. A. Gaunt, “The deferred approach to the limit,” Philos. Trans. Roy. Soc. London, Ser. A, v. 226, 1927, p. 299-361.
  • Gertrude Blanch, On the numerical solution of parabolic partial differential equations, J. Research Nat. Bur. Standards 50 (1953), 343–356. MR 0059078
  • H. C. Bolton and H. I. Scoins, Eigenvalues of differential equations by finite-difference methods, Proc. Cambridge Philos. Soc. 52 (1956), 215–229. MR 79344
  • M. G. Salvadori, Extrapolation formulas in linear difference operators, Proceedings of the First U. S. National Congress of Applied Mechanics, Chicago, 1951, The American Society of Mechanical Engineers, New York, N. Y., 1952, pp. 15–18. MR 0060911
  • Table of sines and cosines to fifteen decimal places at hundredths of a degree, National Bureau of Standards Applied Mathematics Series, No. 5, U. S. Government Printing Office, Washington, D. C., 1949. MR 0030289
  • E. Whittaker & G. Robinson, The Calculus of Observations, 4th edition, Blackie and Son, London, 1954, p. 135. H. T. Davis, Tables of the Higher Mathematical Functions, v. II, Principia Press, Bloomington, Indiana, 1935, p. 282, 284-285, 304. Davis cites the earlier work of J. W. L. Glaisher in relation to Catalan’s constant in Mess. of Math., v. 6, 1876, p. 71-76, Proc. London Math. Soc., v. 8, 1877, p. 200-201, Mess. of Math., v. 42, 1913, p. 35-58. G. N. Watson, Theory of Bessel Functions, 2nd edition, Cambridge University Press, 1952, p. 506. NBS, Tables of Spherical Bessel Functions, v. II, New York, Columbia University Press, 1947, p. 318.
  • W. G. Bickley and J. C. P. Miller, Notes on the evaluation of zeros and turning values of Bessel functions. II. The McMahon series, Philos. Mag. (7) 36 (1945), 124–131. MR 12903

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65.00

Retrieve articles in all journals with MSC: 65.00

Additional Information

Article copyright: © Copyright 1961 American Mathematical Society