Department of Mathematics

Syllabus

**Math 01.331 - Introduction to Real Analysis II**

**COURSE DESCRIPTION:**

Math 01.331 Introduction to Real Analysis II
3 s.h.

(Prerequisite: Math 01.330 Introduction to Real Analysis I with a grade of C- or better)

This course is a continuation of Introduction to Real Analysis I.

The purpose is to extend the student's understanding of basic analysis
and the calculus. Topics included are: the mean-value theorem,
existence of the Riemann integral, Riemann-Stieltjes integration, infinite
series, convergence tests and Fourier series.

**OBJECTIVE:**

Students will demonstrate the ability to use rigorous mathematical thought processes in the following areas: sets, functions, sequences and series, limits, continuity, derivatives, integrals and Fourier series.

**CONTENTS:**

1.0 **Mean Value Theorems**

1.1 Local Maxima and Minima

1.2 Mean Value Theorem

1.3 Taylor's Theorem

2.0 **Riemann Integral**

2.1 Basic definition

2.2 Proof of the existence of the integral

2.3 Fundamental theorem of calculus

2.4 Properties of the integral

2.5 Improper integrals

3.0 **Riemann-Stieltjes Integrals**

3.1 Bounded version

3.2 Basic Theorems

4.0 **Infinite Series**

4.1 Definitions

4.2 Tests for convergence

4.3 Taylor Series

5.0** Sequences and Series of Functions**

5.1 Definitions

5.2 Pointwise and uniform convergence

5.3 Uniform convergence of power series

6.0 **Fourier Series**

6.1 Convergence problems

6.2 Summability of Fourier series

6.3 Convergence of Fourier series

6.4 Orthogonal expansions

**TEXT:**

Generally, the same text is used in Introduction to Real Analysis II
as was used in Introduction to Real Analysis I.