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

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.

- Herbert E. Salzer,
*A simple method for summing certain slowly convergent series*, J. Math. and Phys.**33**(1955), 356–359. MR**68315**, DOI https://doi.org/10.1002/sapm1954331356 - Herbert E. Salzer,
*Formulas for the partial summation of series*, Math. Tables Aids Comput.**10**(1956), 149–156. MR**81528**, DOI https://doi.org/10.1090/S0025-5718-1956-0081528-5
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,” - 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, - 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**

*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.

*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.

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