Rate of Convergence of Positive Series
We investigate the rate of convergence of series of the form $$F(x) = \mathop \sum \limits_{n = 0}^{ + \infty } \;a_n e^{x\lambda _n + \tau (x)\beta _n } ,\quad a_n \geqslant 0,\quad n \geqslant 1,\quad a_0 = 1$$ where λ = (λn), 0 = λ0 < λn ↑ + ∞, n → + ∞, β = {βn: n ≥ 0} ⊂ ℝ+, and τ(x) i...
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| Datum: | 2004 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2004
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3873 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510007115644928 |
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| author | Skaskiv, O. B. Скасків, О. Б. |
| author_facet | Skaskiv, O. B. Скасків, О. Б. |
| author_sort | Skaskiv, O. B. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:13:06Z |
| description | We investigate the rate of convergence of series of the form $$F(x) = \mathop \sum \limits_{n = 0}^{ + \infty } \;a_n e^{x\lambda _n + \tau (x)\beta _n } ,\quad a_n \geqslant 0,\quad n \geqslant 1,\quad a_0 = 1$$ where λ = (λn), 0 = λ0 < λn ↑ + ∞, n → + ∞, β = {βn: n ≥ 0} ⊂ ℝ+, and τ(x) is a nonnegative function nondecreasing on [0; +∞), and $$F(x) = \mathop \sum \limits_{n = 0}^{ + \infty } \;a_n f(x\lambda _n ),\quad a_n \geqslant 0,\quad n \geqslant 1,\quad a_0 = 1,$$ where the sequence λ = (λn) is the same as above and f (x) is a function decreasing on [0; +∞) and such that f (0) = 1 and the function ln f(x) is convex on [0; +∞). |
| first_indexed | 2026-03-24T02:50:09Z |
| format | Article |
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| id | umjimathkievua-article-3873 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:50:09Z |
| publishDate | 2004 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/14/0839ff26611cf4e980dbe531521ab014.pdf |
| spelling | umjimathkievua-article-38732020-03-18T20:13:06Z Rate of Convergence of Positive Series Швидкість збіжності додатних рядів Skaskiv, O. B. Скасків, О. Б. We investigate the rate of convergence of series of the form $$F(x) = \mathop \sum \limits_{n = 0}^{ + \infty } \;a_n e^{x\lambda _n + \tau (x)\beta _n } ,\quad a_n \geqslant 0,\quad n \geqslant 1,\quad a_0 = 1$$ where λ = (λn), 0 = λ0 < λn ↑ + ∞, n → + ∞, β = {βn: n ≥ 0} ⊂ ℝ+, and τ(x) is a nonnegative function nondecreasing on [0; +∞), and $$F(x) = \mathop \sum \limits_{n = 0}^{ + \infty } \;a_n f(x\lambda _n ),\quad a_n \geqslant 0,\quad n \geqslant 1,\quad a_0 = 1,$$ where the sequence λ = (λn) is the same as above and f (x) is a function decreasing on [0; +∞) and such that f (0) = 1 and the function ln f(x) is convex on [0; +∞). Досліджується швидкість збіжності ряді» вигляду $$F(x) = \mathop \sum \limits_{n = 0}^{ + \infty } \;a_n e^{x\lambda _n + \tau (x)\beta _n } ,\quad a_n \geqslant 0,\quad n \geqslant 1,\quad a_0 = 1$$ де $λ = (λ_n),\; 0 = λ_0 < λ_n ↑ + ∞,\; n → + ∞, \;β = {β_n: n ≥ 0} ⊂ ℝ_{+}$, а $τ(x)$ — невід'ємна неспадна на $[0; +∞)$ функція; $$F(x) = \mathop \sum \limits_{n = 0}^{ + \infty } \;a_n f(x\lambda _n ),\quad a_n \geqslant 0,\quad n \geqslant 1$$ Тут послідовність $λ = (λ_n)$ така ж, як і вище, a $f (x)$—додатна зростаюча на $[0; +∞)$ функція така, що $f (0) = 1$, а функція $\ln f(x)$ — опукла на $[0; +∞)$. Institute of Mathematics, NAS of Ukraine 2004-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3873 Ukrains’kyi Matematychnyi Zhurnal; Vol. 56 No. 12 (2004); 1665-1674 Український математичний журнал; Том 56 № 12 (2004); 1665-1674 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3873/4464 https://umj.imath.kiev.ua/index.php/umj/article/view/3873/4465 Copyright (c) 2004 Skaskiv O. B. |
| spellingShingle | Skaskiv, O. B. Скасків, О. Б. Rate of Convergence of Positive Series |
| title | Rate of Convergence of Positive Series |
| title_alt | Швидкість збіжності додатних рядів |
| title_full | Rate of Convergence of Positive Series |
| title_fullStr | Rate of Convergence of Positive Series |
| title_full_unstemmed | Rate of Convergence of Positive Series |
| title_short | Rate of Convergence of Positive Series |
| title_sort | rate of convergence of positive series |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3873 |
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