## Math-451 Real Analysis II

### Mathematics, Statistics & Computer Science Department

COURSE NO./TITLE: MATH-451 (355-451) REAL ANALYSIS II

CREDIT: 3

COURSE DESCRIPTION: Continuation of MATH-450; differentiation, integration, infinite series.

Prerequisite: MATH-450.

1. TEXTBOOK: Introduction to Real Analysis, 2nd Ed., by Stoll (adopted 8/01) & Closer & Closer – Introducing Real Analysis, 1st Ed., by Schumacher (adopted F08)
2. (Previously used: Advanced Calculus, 1st Ed., by Fitzpatrick.)
3. (Prior to Spring 98: Introduction to Real Analysis, 2nd Ed., by Bartle and Sherbert)

COURSE OBJECTIVES:

1. To review elementary mathematics from an advanced standpoint.
2. To provide insight into the role of rigor in mathematics.
3. To build understanding of the foundations and concepts of analysis.
4. To strengthen powers of logic (reasoning).
5. To provide appreciation of and insight into the applications of analysis to statistics and numerical analysis.

COURSE OUTLINE:

1. Differentiation
- Rolle's Theorem
- Mean Value Theorem
- Interchange of Limit and Derivative
- Directional Derivative
- Chain Rule
2. Integration
- Riemann-Stieltjes Integral (Existence and Properties)
- First Mean Value Theorem
- Fundamental Theorem of Integral Calculus
- Integration by Parts
- Second Mean Value Theorem
3. Infinite Series
-Convergence (Cauchy Criterion, Absolute, Rearrangement of Series)
-Tests of Convergence (Comparison, Root, Ratio, Integral, Alternating Series)
4. Series of Functions
- Absolute and Uniform Convergence
- Continuity of the Limit
- Tests for Uniform Convergence

Revised 6/08