20.500.12556/DKUM-38626 Verižni ulomki in vrste Continued fractions and series V tem diplomskem delu preučujemo zvezo med navadnimi verižnimi ulomki in neskončnimi številskimi vrstami. V prvem poglavju opišemo osnove teorije navadnih verižnih ulomkov. Posebej obravnavamo končne, neskončne in periodične verižne ulomke. Verižni ulomki so uporabni pri iskanju najboljših racionalnih aproksimacij iracionalnih števil. Mnogi matematiki so se v preteklosti ukvarjali s problemom, kako povezati verižne ulomke z vrstami. V drugem poglavju izpeljemo pomembne rezultate, ki povezujejo navadne verižne ulomke in vrste. Prvi pomembni rezultat je izrek, s katerim kvadratično iracionalno število razvijemo v vrsto, katere delne vsote so konvergenti ustreznega navadnega verižnega ulomka. Obratno lahko ta izrek uporabimo za iskanje navadnih verižnih ulomkov vsot nekaterih tipov vrst. V zadnjem poglavju obravnavamo Newtonovo metodo, s katero dobimo zaporedne približke kvadratično iracionalnih števil in jih primerjamo s približki kvadratično iracionalnih števil pridobljenimi s konvergenti navadnega verižnega ulomka. The graduation thesis deals with a connection between continued fractions and infinite series. In the first chapter, the basics of the theory of simple continued fractions are presented. We consider finite, infinite and periodic continued fractions. Continued fractions are useful when searching the best rational approximations of irrational numbers. In the past, many mathematicians tried to resolve the issue how to connect continued fractions with series. The second part of the graduation thesis shows significant results which connect continued fractions and series. The first important result is a theorem which helps us expand a quadratic irrational number into series, where its partial sums are convergents of a corresponding simple continued fraction. Inversely, this theorem can be used for searching simple continued fractions of sums of some types of series. And the last chapter deals with the Newton's method which provides us with successive approximations of quadratic irrational numbers and which are compared with approximations of quadratic irrational numbers obtained with convergents of a simple continued fraction. verižni ulomki končni verižni ulomki neskončni verižni ulomki periodični verižni ulomki vrste Newtonova metoda kvadratično iracionalno število aproksimacija konvergent zaporedje continued fractions finite continued fractions infinite continued fractions periodic continued fractions series Newton's method a quadratic irrational number approximation convergent sequence. true true false [S. Ferk] Slovenski jezik Angleški jezik Diplomsko delo 2012-10-05 13:59:38 2012-11-13 14:41:29 2022-05-04 13:28:34 2021-04-20 03:21:06 2012 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 511.524(043.2) 19472648 URN:SI:UM:DK:QDIWLSFA UNI_Ferk_Sonja_2012.pdf UNI_Ferk_Sonja_2012.pdf 1 9B420AA2FA5276BEF33818FFAAE8B8E1 e58092df4ce792a07ed48b5767feb50a5136087b4f4f75a0127e09f4dddf8491 864d18b4-7c08-11eb-bb7a-00155d0001ca 20.500.12556/dkum/ca3d03b8-7f88-4c2c-83bb-9754f6ee2412 https://dk.um.si/Dokument.php?lang=slv&id=51760 Fakulteta za naravoslovje in matematiko 0 0 0