Publications de l'Institut Mathematique 2022 Volume 111, Issue 125, Pages: 41-53
https://doi.org/10.2298/PIM2225041A
Full text (
239 KB)
Evaluating the sum of convergent positive series
Abramov Vyachslav M., vabramov126@gmail.com
We provide numerical procedures for possibly best evaluating the sum of
positive series under quite general setting. Our procedures are based on the
application of a generalized version of Kummer's test.
Keywords: Kummer's test, numerical algorithms, positive series, Tauberian theorems, Tong's theorem
Show references
V. M. Abramov, Extension of the Bertrand-De Morgan test and its application, Amer. Math. Monthly, 127 (2020), 444-448.
D. H. Bailey, The SIAM 100-Digit Challenge: A study in high-accuracy numerical computing, by Folkmar Bornemann, Dirk Laurie, Stan Wagon, and Jörg Waldvogel, SIAM, Philadelphia, PA, 2004. (Book review.) Bull. Amer. Math. Soc., n.s., 42 (2005), 545-548.
R. P. Boas, Jr., Partial sums of infinite series, and how they grow. Amer. Math. Monthly, 84 (1977), 237-258.
F. Bornemann, D. Laurie, S. Wagon, J. Waldvogel. The SIAM 100-Digit Challenge: A Study in High-Accuracy Numerical Computing, SIAM, Philadelphia, 2004.
J. M. Borwein, The SIAM 100-Digit challenge: a study in high-accuracy numerical computing, Math. Intelligen., 27 (2005), 40-48.
B. Braden, Calculating sums of infinite series, Amer. Math. Monthly, 99 (1992): 649-655.
R. J. Collings, Approximating series, College Math. J., 23 (1992), 153-157.
W. Gautschi, G. V. Milovanović, Gaussian quadrature involving Einstein and Fermi functions with an application to summation of series, Math. Comp. 44 (1985), 177-190; Supplement to Gaussian quadrature involving Einstein and Fermi functions with an application to summation of series, Math. Comp. 44 (1985), S1-S11.
I. Győri, L. Horváth, lp-solutions and stability analysis of difference equations using the Kummer's test, Appl. Math. Comput., 217 (24) (2011), 10129-10145.
G. H. Hardy, Divergent Series, 2nd ed. Providence: AMS Chelsea Publishing, 2000.
G. H. Hardy, J. E. Littlewood, Tauberian theorems concerning power series and Direchlet's series whose coefficients are positive, Proc. London Math. Soc. 13 (1914) 174-191.
N. J. Higham, The SIAM 100-Digit Challenge: A study in high-accuracy numerical computing, by Folkmar Bornemann, Dirk Laurie, Stan Wagon, and Jörg Waldvogel, SIAM, Philadelphia, PA, 2004. (Book review.) SIAM Rev., 47 (2005), 382-383.
K. Knopp, Theory and Application of Infinite Series, New York: Dover Publ. Inc., 1990.
E. Kummer, Über die Convergenz und Divergenz der unendlichen Reihen, J. für die Reine und Angew. Math. 13 (1835), 171-184.
G. V. Milovanović, Summation of series and Gaussian quadratures, In: Approximation and Computation (R.V.M. Zahar, ed.), pp. 459-475, ISNM Vol. 119, Birkhäuser, Basel-Boston-Berlin, 1994.
G. V. Milovanović, Methods for computation of slowly convergent series and finite sums based on Gauss-Christoffel quadratures, Jaen. J. Approx. 6 (2014), 37-68.
G. V. Milovanović, On summation/integration methods for slowly convergent series, Stud. Univ. Babeş-Bolyai Math. 61 (2016), 359-375.
G. V. Milovanović, Summation formulas of Euler-Maclaurin and Abel-Plana: old and new results and applications, In: Progress in Approximation Theory and Applicable Complex Analysis - In the Memory of Q.I. Rahman (N.K. Govil, R.N. Mohapatra, M.A. Qazi, G. Schmeisser, eds.), pp. 429-461, Springer, 2017.
G. V. Milovanović, Quadrature formulas of Gaussian type for fast summation of trigonometric series, Constr. Math. Anal. 2, No. 4 (2019), 168-182.
G. V. Milovanović, Summation of slowly convergent series by the Gaussian type of quadratures and application to the calculation of the values of the Riemann zeta function, Bull. Cl. Sci. Math. Nat. Sci. Math. 46 (2021), 131-150.
R. K. Morley, The remainder in computing by series, Amer. Math. Monthly 57 (1950), 550-551.
R. B. Nelsen, An improved remainder estimate for use with the integral test, College Math. J. 34 (2003), 397-399.
P. Řehák, Kummer test and regular variation, Monatsh. Math., 192 (2020), 419-426.
J. Tong, Kummer's test gives a characterization of convergence or divergence for all positive series, Amer. Math. Monthly, 101 (1994), 450-452.
L. N. Trefethen, The $100, 100-Digit Challenge, SIAM News, 35 (2002), 1-3.
