Preureditve pogojno konvergentnih številskih vrst

magistrsko delo

Barbara Vlah (Author), Tadej Starčič (Mentor)

Abstract

V magistrskem delu bomo obravnavali pogojno konvergentne neskončne številske vrste. Zanimalo nas bo, kako in kdaj vrstni red seštevanja členov take številske vrste vpliva na samo vsoto. Za pogojno konvergentne vrste z realnimi členi velja Riemannov izrek, ki nam pove, da je lahko pri ustrezni preureditvi vsota vrste poljubno število. Pogledali si bomo nekaj konkretnih preureditev in pripadajočih vsot alternirajoče harmonične številske vrste ter Schlömilchov in Pringsheimov izrek za alternirajoče številske vrste. V kompleksnem primeru pa bomo preštudirali zahtevnejši Lévy–Steinitzov izrek, ki pravi, da so možne vsote bodisi števila na neki premici v kompleksni ravnini bodisi celotna kompleksna ravnina.

Keywords

številska vrsta;absolutna konvergenca;pogojna konvergenca;preureditev vrste;Riemannov izrek;Schlömilchov izrek;Pringsheimov izrek;Lévy–Steinitzov izrek;

Data

Language:	Slovenian
Year of publishing:	2025
Typology:	2.09 - Master's Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[B. Vlah]
UDC:	51(043.2)
COBISS:	236282371
Views:	25
Downloads:	4
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Rearrangements of conditionally convergent series
Secondary abstract:	In the master's thesis we will discuss conditionally convergent infinite number series. We will look at how and when the order of the summation terms of such a number series affects the sum itself. For conditionally convergent series with real terms, Riemann series theorem tells us that with appropriate rearrangement the sum of the series can be an arbitrary number. We will examine some specific rearrangements and corresponding sums of alternating harmonic series, the Schlömilch theorem and the Pringsheim theorem for alternating series. In a complex case, we will study the more challenging Lévy–Steinitz theorem, which says that the set of all possible sums is either a line in a complex plane or the entire complex plane.
Secondary keywords:	number series;absolute covergence;conditional convergence;rearrangement of series;Riemann series theorem;Schlömilch theorem;Pringsheim theorem;Lévy–Steinitz theorem;Matematika;Univerzitetna in visokošolska dela;
Type (COBISS):	Master's thesis/paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Pedagoška fak., Poučevanje
Pages:	1 spletni vir (1 datoteka PDF (57 str.))
ID:	26394071