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.