Functions that preserve the convergence of a series

V magistrskem delu vpeljemo funkcije, ki ohranjajo konvergenco vrst. To so funkcije newline $f:RR longrightarrow RR$, za katere velja, da iz konvergence vrste $sumlimits_{n=1}^{infty}a_n$ sledi konvergenca vrste $sumlimits_{n=1}^{infty}f(a_n)$. Glavni cilj magistrske naloge je podati in dokazati enostavno karakterizacijo takih funkcij. Najprej bomo predstavili osnovne pojme iz Analize in Algebre, ki so potrebni za razumevanje rezultatov, ki jih opisujemo v magistrskem delu. V drugem poglavju podamo in dokažemo glavni izrek magistrskega dela o karakterizaciji funkcij, ki ohranjajo konvergenco vrst. V zadnjem poglavju preučujemo posplošitve takih funkcij. Natančneje, definiramo lastnost, kot so biti $(c, ac)$, $(ac,c)$, $(acp)$ funkcija, in dokažemo zanimive rezultate za opisane razrede funkcij.This Master’s thesis introduces a family of function that preserve the convergence properties of a series. Namely these are functions $f:RR longrightarrow RR$: such that if $sumlimits_{n=1}^{infty}a_n$ converges then $sumlimits_{n=1}^{infty}f(a_n)$ also converges. The main aim of the thesis is to mathematically determine a simple characterisation of such functions. First, some background from Analysis and Algebra is introduced. The second chapter provides and proves the main theorem that characterises functions that preserve the convergence of a series. The final chapter investigates a generalisation of such functions. Specifically, some special kinds of functions are defined, such as $(c, ac)$, $(ac,c)$, $(acp)$ functions, which leads to interesting results for these classes of functions