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 , 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.

**[1]**Herbert E. Salzer,*A simple method for summing certain slowly convergent series*, J. Math. and Phys.**33**(1955), 356–359. MR**0068315****[2]**Herbert E. Salzer,*Formulas for the partial summation of series*, Math. Tables Aids Comput.**10**(1956), 149–156. MR**0081528**, 10.1090/S0025-5718-1956-0081528-5**[3]**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.**[4]**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.**[5]**Gertrude Blanch,*On the numerical solution of parabolic partial differential equations*, J. Research Nat. Bur. Standards**50**(1953), 343–356. MR**0059078****[6]**H. C. Bolton and H. I. Scoins,*Eigenvalues of differential equations by finite-difference methods*, Proc. Cambridge Philos. Soc.**52**(1956), 215–229. MR**0079344****[7]**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****[8]***Table of sines and cosines to fifteen decimal places at hundredths of a degree*, Appl. Math. Ser., no. 5, National Bureau of Standards., 1949. MR**0030289****[9]**E. Whittaker & G. Robinson,*The Calculus of Observations*, 4th edition, Blackie and Son, London, 1954, p. 135.**[10]**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.**[11]**G. N. Watson,*Theory of Bessel Functions*, 2nd edition, Cambridge University Press, 1952, p. 506.**[12]**NBS,*Tables of Spherical Bessel Functions*, v. II, New York, Columbia University Press, 1947, p. 318.**[13]**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**0012903**

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

Article copyright:
© Copyright 1961
American Mathematical Society