Analysis Mathematica

, Volume 6, Issue 1, pp 3–50 | Cite as

Взаимоотношение меж ду сходимостью подпоследовательно стей частных сумм числово го ряда и его суммируе мостью методами (C, α) и Абеля

  • Д. Е. Меняшов
Article

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

Abstract

Numerical series\(\mathop \Sigma \limits_{n = 0}^\infty u_n\) with partial sumss n are studied under the assumption that a subsequence\(\left\{ {S_{n_k } } \right\}_{k = 0}^\infty\) 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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Литература

  1. [1]
    Д. Е. Меньшов, Взаим оотношение между схо димостью подпоследо вательностей частны х сумм числового ряда и его суммируемостью методом Абеля,Докл. А Н СССР,235 (1977), 27–29.Google Scholar

Copyright information

© Akadémiai Kiadó 1980

Authors and Affiliations

  • Д. Е. Меняшов
    • 1
  1. 1.МАТЕМАТИЧЕСКИЙ ИНСТ ИТУТИМ. В. А. СТЕКЛОВА А Н СССР ЛитератураСССР, МОСК ВА

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