Generalized Harmonic Series on Hyperreal Structures in the AST
Abstract
A most useful applications the Alternative Set Theory gets in studying several classical problems of calculus from a fresh point of view. Here, the class of all real numbers is replaced by a hyperreal structure based on some initial segment of natural numbers class. This segment must have more or less degree of uncertainty. It represents some horizon and is called a cut. This hyperreal structure inherits some properties of the real numbers, and some not. We investigate the relation of inherited properties with the uncertainty degree of the main cut of the structure. In this paper, an attempt to extrapolate the theory of numerical series summing to the hyperreal structures is presented. In particular, the classical theorem of comparison for numerical series is verified. The main result of the research is related with the so-called generalized harmonic series - the series of the type ∑ n-p. We show that if the main cut is closed with respect to the multiplication of its elements, then, roughly speaking, the series converges for p > 1. If the cut is additive but not multiplicative then it converges for p > 2 only, which is rather surprising. This can be used to characterize a degree of uncertainty of the cut by the minimal p for which the series of the investigated type converges on the correspondent hyperreal structure.Downloads
Metrics
References
Stewart J. Calculus: Early Transcendentals. - N.Y., 2014.
Хренников А.Ю. Суперанализ. - М., 2014.
Вопенка П. Альтернативная теория множеств. Новый взгляд на бесконечность. - Новосибирск, 2004.
Вопенка П. Математика в альтернативной теории множеств. - М., 1983.
Kalina M., Zlatos P. Arithmetic of cuts and cuts of classes // Comment. Math. Univ. Carolinae. -1988. - № 29.
Kalina M., Zlatos P. Cuts of real classes. // Comment. Math. Univ. Carolinae. - 1989. - № 30.
Sochor A. Addition of initial segments I, II. // Comment. Math. Univ. Carolinae. - 1988. - № 29.
Дронов С.В. О пределах монотонных последовательностей в AST. // Известия Алтайского гос. ун-та. - 2016. - №1(89). DOI:10.14258/izvasu(2016)1-19.
Фихтенгольц Г.М. Курс дифференциального и интегрального исчисления : в 3 т. - М., 2016.
Дронов С.В. О свойстве фундаментальности сегментов класса натуральных чисел // Известия Алтайского гос. ун-та. - 2009. - № 1/1.
Izvestiya of Altai State University is a golden publisher, as we allow self-archiving, but most importantly we are fully transparent about your rights.
Authors may present and discuss their findings ahead of publication: at biological or scientific conferences, on preprint servers, in public databases, and in blogs, wikis, tweets, and other informal communication channels.
Izvestiya of Altai State University allows authors to deposit manuscripts (currently under review or those for intended submission to Izvestiya of Altai State University) in non-commercial, pre-print servers such as ArXiv.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC BY 4.0) that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
