ON SOME CONVERGENCE TESTS FOR ALTERNATING SERIES AND CONSTANT SIGN SERIES
https://doi.org/10.22405/2226-8383-2017-18-1-123-133
Abstract
Well known properties of numerical series P+1 n=1 an in the course of analysis, which have asymptotic growth of powers of n at innity. Relevant tests of convergence was laid in the works of Gauss. We study the necessary and sucient conditions for the positive (and constant sign) a sequence of numbers fang+1 n=1 with the rate of decrease (growth) in logarithmic scale for the convergence of the series P+1 n=1 an. 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 nding the spectrum of the operator of SturmLiouville 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 SturmLiouville on the half-line.
About the Author
A. I. Kozko
МГУ имени М.В. Ломоносова
РАН ХиГС
Russian Federation
Russian Federation
Candidate of Physico-Mathematical Sciences, Docent, MSU Faculty of Mechanics and Mathematics, RANEPA
References
1. Ilyin V. A. Sadovnichii V. A. Sendov Bl. H. 1985, ``Mathematical Analysis``, M. Izdatelstvo MSU, vol. 1, 662 pp. vol. 2, 358 pp.
2. Nikolskii S. M. 2001,``Course of mathematical analysis``, M. Fizmatlit, 592 p.
3. Shvedov I. A. 2003,``Kompaktnyi kurs matematicheskogo analiza``, Novosib. Gos. Univ., 112 pp.
4. Zorich Vladimir A. 2015,``Mathematical Analysis I``, 2016, ``Mathematical Analysis II``, Springer.
5. Kozko A. I. 2005, ``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, pp. 636–648.
6. Kozko A. I. 2016, ``The properties of the eigenvalues of Sturm-Liouville in $L^2(\mathbbR_+)$ with boundary condition $y(0)\cos \alpha +y'(0) \sin \alpha =0$``, Sbornik 18 Mezdunarodnoi Saratovskoi zimnei shkolu «Sovremennye problemy teorii funkchii i ih prilozeniya» (27 january–03 february 2016), Izdatelstvo SGU Saratov, pp. 150--152.
7. Pechentsov A. S. 1999, ``Regularized traces of differential operators: the Lidskii–Sadovnichii method``, Differential Equations, 35:4, pp. 490–497. %``Следы одного класса сингулярных дифференциальных операторов:метод Лидского-Садовничего`` //%Вестник Московского университета. Серия 1. Математика и механика. 1999. № 5, С. 35--42.
8. Kozko A. I., Pechentsov A. S. 2011,``Regularized traces of singular differential operators with canonical boundary conditions``, Moscow Univ. Math. Bulletin, vol. 66, no. 4.
9. Kozko A. I., Pechentsov A. S. 2008, ``Regularized traces of higher-order singular differential operators``, Mathematical Notes, 83:1, pp. 37 DOI: https://doi.org/10.4213/mzm3764
10. Sadovnichii V. A., Pechentsov A. S., Kozko A. I. 2009, ``Regularized traces of singular differential operators``, Doklady Mathematics august, vol. 80, no. 1, pp. 550--554. №indent DOI: 10.1134/S1064562409040255
11. Kozko A. I., Pechentsov A. S. 2005, ``Spectral functions and regularized traces of singular differential operators of higher orders``, Doklady Mathematics, vol. 71, no. 2, pp. 195-197.
12. Kozko A. I. 2004, ``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. pp. 1-22.
13. Kozko A. I., Pechentsov A.S. 2004, ``On the spectral function and regularized traces of singular differential operators``, International conference "m-Function and Related Topics Conference" dedicated to professor W. N. Everitt, Cardiff, Wales, Uk, July 19 - 21, pp. 14-16.
14. Kozko A. I., Pechentsov A. S. 2004, ``Regularized traces of singular differential operators of higher orders``, IWOTA, Newcastle upon Tyne, Uk, July 12 - 16, pp. 31-31
15. Kozko A. I., Pechentsov A. S. 2010, ``The spectral function of a singular differential operator of order $2m$``, Izvestiya: Mathematics, 74:6, pp. 1205–1224. DOI: https://doi.org/10.4213/im2780
16. Pechentsov A. S. 1998, ``Regularized traces of boundary problems in case of multiple roots of characteristic polynomial``, Fundamtntalnaya i priklodnaya matematika, vol -4, no. 2, pp. 567--583.
17. Kadchenko S. I. 2009, ``Method of regularized traces``, Bulletin of the South Ural State University. Series: Mathematical modelling, programming and computer software, no. 37(170), issue 4, pp. 4--23.
18. Raspopov V. V., Dubrovskii V. V. 2002, ``An approximate regularized trace formula for an ordinary fourth-order differential operator``, Differential Equations, 38:7, pp. 1042--1045.
19. Bobrov A. N., Podol'skii V. E. 1999, ``Convergence of regularized traces of powers of the Laplace-Beltrami operator with potential on the sphere $S^n$``, Sbornik: Mathematics, 190:10, pp. 1401–1415. DOI: https://doi.org/10.4213/sm430
20. Sadovnichii V. A., Podolskii V. E. 2006, ``Regularized traces of discrete operators`` // Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 255, suppl. 2, pp. 161--177.
21. Sadovnichii V. A., Podolskii V. E. 2006, ``Traces of operators ``, Uspekhi Mat. Nauk, vol. 61, no. 5(371), pp. 89--156. DOI: https://doi.org/10.4213/rm4833
Review
For citations:
Kozko A.I. ON SOME CONVERGENCE TESTS FOR ALTERNATING SERIES AND CONSTANT SIGN SERIES. Chebyshevskii Sbornik. 2017;18(1):123-133. (In Russ.) https://doi.org/10.22405/2226-8383-2017-18-1-123-133
Views:
758
Email this article 