Real Analysis

(3-4.5 Credit Hours) · Undergraduate

The follwoing more robust course description was provided for the version of the course taught at UCD in Dublin. "Analysis" is a module designed to introduce to students some of the theory developed on sequences and series in the 19th century. The student will be introduced to the concept of a sequence of real numbers and will learn about various other concepts such as that of convergent sequence, bounded sequence and monotonic sequence to name a few. The Axiom of Completeness will be introduced and the student will see its role in proving results such as the Monotone Convergence Theorem and Bolzano-Weierstrass Theorem. The concepts of countable and uncountable sets will also be discussed and the student will learn that the set of real numbers is uncountable. The latter part of the module deals with tests of convergence for series and looks at the concepts of conditional and absolute convergence. The module concludes with a discussion of power series and Taylor series. On completion of this module the student should be able to:-Define and give examples of the major concepts of sequences of real numbers.-Describe and give examples of the relationships between the major concepts of sequences of real numbers.-Prove various results concerning sequences of real numbers.-Describe and apply the Axiom of Completeness.-Define and describe the concepts of countable and uncountable sets.-Test a given series for convergence.-Find the interval of convergence of a power series.-Find the Taylor series generated by a given function. Pre-Requisite : Intro to Analysis (E&F) (MATH10130), Advanced Calculus (E&F) (MATH10140), Calculus I (MST10010), Calculus II (MST10020)