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# Interrelations between convergence of subsequences of partial sums of numerical series and its summability by (C, α) and Abel means

Analysis Mathematica
Author: Д. Меняшов

## Abstract

Numerical series
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop \Sigma \limits_{n = 0}^\infty u_n$$ \end{document}
with partial sumss n are studied under the assumption that a subsequence
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\{ {S_{n_k } } \right\}_{k = 0}^\infty$$ \end{document}
of the partial sums is convergent. Then a sequence {η k} is chosen, by means of which a majorant of the termsu n is constructed. Conditions on {n k} and {η k} are found which imply the (C, 1)-summability of the series∑ u n (Theorem 1). In the meanwhile, it is proved that the (C, 1)-means in Theorem 1 cannot be replaced by (C, α)-means, if 0<α<1 (Theorem 2). On the other hand, if the assumption in Theorem 1 is not satisfied, then in certain cases the series∑ u n preserves the property of (C, 1)-summability (Theorems 4 and 5), while in other cases it is not summable even by Abel means (Theorems 3 and 6).
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