Some transformations of monotone sequences
Abstract: In this paper, we discuss a class of methods for summing sequences which are generalizations of a method due to Salzer. The methods are not regular, and in contrast to the classical regular methods, seem to work best on sequences which are monotone. In our main theorem, we determine a class of convergent sequences for which the methods yield sequences which converge to the same sum.
-  Herbert E. Salzer, A simple method for summing certain slowly convergent series, J. Math. and Phys. 33 (1955), 356–359. MR 0068315, 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 0081528, https://doi.org/10.1090/S0025-5718-1956-0081528-5
-  Herbert E. Salzer and Genevieve M. Kimbro, Improved formulas for complete and partial summation of certain series., Math. Comp. 15 (1961), 23–39. MR 0121972, https://doi.org/10.1090/S0025-5718-1961-0121972-2
-  N. E. Nörlund, Vorlesungen über die Differenzenrechnung, Springer, Berlin, 1924; reprint, Chelsea, New York, 1954.
- H. E. Salzer, ``A simple method for summing certain slowly convergent series,'' J. Math. Physics, v. 33, 1955, pp. 356-359. MR 16, 863. MR 0068315 (16:863d)
- H. E. Salzer, ``Formulas for the partial summation of series,'' MTAC, v. 10, 1956, pp. 149-156. MR 18, 416. MR 0081528 (18:416a)
- H. E. Salzer & G. M. Kimbro, ``Improved formulas for complete and partial summation of certain series,'' Math. Comp., v. 15, 1961, pp. 23-39. MR 22 #12699. MR 0121972 (22:12699)
- N. E. Nörlund, Vorlesungen über die Differenzenrechnung, Springer, Berlin, 1924; reprint, Chelsea, New York, 1954.
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Keywords: Monotone sequences, summation methods, psi function
Article copyright: © Copyright 1972 American Mathematical Society