On some convergence tests for alternating series and constant sign series

A. I. Kozko

Čebyševskij sbornik, Tome 18 (2017) no. 1, pp. 123-133.

Résumé

Well known properties of numerical series $\sum_{n=1}^{+\infty}a_n$ in the course of analysis, which have asymptotic growth of powers of $n$ at infinity. Relevant tests of convergence was laid in the works of Gauss. We study the necessary and sufficient conditions for the positive (and constant sign) a sequence of numbers $\{a_n\}_{n=1}^{+\infty}$ with the rate of decrease (growth) in logarithmic scale for the convergence of the series $\sum_{n=1}^{+\infty}a_n$. Examples of the use of the criteria of convergence, as in the case of constant sign of series, and in the case of alternating series. The importance of a logarithmic scale due to the fact that it is found in various sections of the analysis and, in particular, the problem of finding the spectrum of the operator of Sturm–Liouville on the half-line for the fast growing potentials. On a logarithmic scale arise and the relevant questions on the presence of regularized sums, for the special potentials of the operator of Sturm–Liouville on the half-line. This work is devoted to the seventieth Doctor of Physical and Mathematical Sciences, Professor Vasily Ivanovich Bernik. In her curriculum vitae, a brief analysis of his scientific work and educational and organizational activities. The work included a list of 80 major scientific works of V. I. Bernik. Bibliography: 21 titles.

Détail

Keywords: convergence of the series, series of constant sign, alternating series, test of convergence of series, test asymptotic behavior, asymptotic expansion, the spectrum of the operator of Sturm–Liouville.

Bibliographie

[1] Ilyin V. A., Sadovnichii V. A., Sendov Bl. H., Mathematical Analysis, v. 1, Izdatelstvo MSU, M., 1985, 662 pp.; v. 2, 1987, 358 pp.

[2] Nikolskii S. M., Course of mathematical analysis, Fizmatlit, M., 2001, 592 pp.

[3] Shvedov I. A., Kompaktnyi kurs matematicheskogo analiza, v. 1, Novosib. Gos. Univ., 2003, 112 pp.

[4] Zorich V. A., Matematicheskii analiz, v 2-kh chastyakh, izd. 4-e, MTsNMO, 2002; Zorich V. A., Mathematical Analysis, v. I, Springer, 2015 ; Mathematical Analysis, v. II, 2016 | MR | Zbl

[5] Kozko A. I., “Asymptotics of the spectrum of the differential operator $-y''+q(x)y$ with a Boundary Condition at zero and with rapidly growing potential”, Differential Equations, 41:5 (2005), 636–648 | DOI | MR | Zbl

[6] Kozko A. I., “The properties of the eigenvalues of Sturm-Liouville in $L^2(\mathbb{R}_+)$ with boundary condition $y(0)\cos \alpha +y'(0) \sin \alpha =0$”, Sovremennye problemy teorii funkchii i ih prilozeniya, Sbornik 18 Mezdunarodnoi Saratovskoi zimnei shkolu (27 january–03 february 2016), Izdatelstvo SGU, Saratov, 2016, 150–152

[7] Pechentsov A. S., “Regularized traces of differential operators: the Lidskii–Sadovnichii method”, Differential Equations, 35:4 (1999), 490–497 | Zbl

[8] Kozko A. I., Pechentsov A. S., “Regularized traces of singular differential operators with canonical boundary conditions”, Moscow Univ. Math. Bulletin, 2011, no. 4, 11–17

[9] Kozko A. I., Pechentsov A. S., “Regularized traces of higher-order singular differential operators”, Mathematical Notes, 83:1 (2008), 37 | DOI | DOI | MR | Zbl

[10] Sadovnichii V. A., Pechentsov A. S., Kozko A. I., “Regularized traces of singular differential operators”, Doklady Mathematics, 80:1 (2009), 550–554 | DOI | MR | Zbl

[11] Kozko A. I., Pechentsov A. S., “Spectral functions and regularized traces of singular differential operators of higher orders”, Doklady Mathematics, 71:2 (2005), 195–197 | MR | Zbl

[12] Kozko A. I., On the spectral functions for the higher-order differential operators with trivial boundary conditions and regularized trace of perturbation operators of the first order, Laboratoire de Mathematiques, Universite Blaise Pascal–Clermont-Ferrand, France, 2004, 22

[13] Kozko A. I., Pechentsov A. S., “On the spectral function and regularized traces of singular differential operators”, m-Function and Related Topics Conference, International conference, dedicated to professor W. N. Everitt (Cardiff, Wales, Uk, July 19–21, 2004), 14–16

[14] Kozko A. I., Pechentsov A. S., “Regularized traces of singular differential operators of higher orders”, IWOTA 2004 (Newcastle upon Tyne, Uk, July 12–16), 31

[15] Kozko A. I., Pechentsov A. S., “The spectral function of a singular differential operator of order $2m$”, Izvestiya: Mathematics, 74:6 (2010), 1205–1224 | DOI | DOI | MR | Zbl

[16] Pechentsov A. S., “Regularized traces of boundary problems in case of multiple roots of characteristic polynomial”, Fundamentalnaya i prikladnaya matematika, 4:2 (1998), 567–583 | Zbl

[17] Kadchenko S. I., “Method of regularized traces”, Bulletin of the South Ural State University. Series: Mathematical modelling, programming and computer software, 37(170):4 (2009), 4–23

[18] Raspopov V. V., Dubrovskii V. V., “An approximate regularized trace formula for an ordinary fourth-order differential operator”, Differential Equations, 38:7 (2002), 1042–1045 | DOI | MR | Zbl

[19] Bobrov A. N., Podol'skii V. E., “Convergence of regularized traces of powers of the Laplace–Beltrami operator with potential on the sphere $S^n$”, Sbornik: Mathematics, 190:10 (1999), 1401–1415 | DOI | DOI | MR | Zbl

[20] Sadovnichii V. A., Podolskii V. E., “Regularized traces of discrete operators”, Proceedings of the Steklov Institute of Mathematics, 255:2, Supplementary issues (2006), 161–177 | MR | Zbl

[21] Sadovnichii V. A., Podolskii V. E., “Traces of operators”, Uspekhi Mat. Nauk, 61:5(371) (2006), 89–156 | DOI | Zbl