Abstract
On the basis of the concepts of (C) -point and (¯R, p)-point of a sequence of complex numbers introduced by the author and results established earlier, we formulate necessary and sufficient conditions for the summability of a number series by a positive Cesaro method or the Riesz method to imply the convergence of this series. We also present a sufficient condition for summability to imply the convergence of a subsequence of its partial sums.
Similar content being viewed by others
References
N. A. Davydov, “One property of Cesaro methods for the summation of series,”Mat. St.,38, 509–524 (1956).
N. A. Davydov, “(C)-property of Cesaro and Abel-Poisson methods and Tauberian-type theorems,”Mat. St.,60, No. 2, 185–206 (1963).
G. A. Mikhalin and L. S. Teslenko, “One property of a class of (¯R, p n ) -methods for the summation of series and Tauberian-type theorems,”Ukr. Mat. Zh.,29, No. 2, 194–203 (1977).
G. Hardy,Divergent Series [Russian translation], Inostr. Lit., Moscow (1951).
M. A. Evgrafov, “On the inversion of the Abel theorem for series with gaps,”Izv. Akad. Nauk SSSR. Ser. Mat.,16, No. 6, 521–524 (1952).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 6, pp. 747–754, June, 1995.
Rights and permissions
About this article
Cite this article
Davydov, N.A. On necessary and sufficient conditions for the summability of a numerical series to imply its convergence. Ukr Math J 47, 860–868 (1995). https://doi.org/10.1007/BF01058776
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01058776