Precise description of the real number system; rigorous treatment of limits and convergence for sequences, series, and functions; continuity and the maximum and intermediate value theorems; differentiation and the mean value theorem; Taylor approximation.
Athena Title
Intro Mathematical Analysis
Equivalent Courses
Not open to students with credit in MATH 3100H, MATH 3100W
Prerequisite
MATH 2260 or MATH 2260E or MATH 2260H or MATH 2310H
Semester Course Offered
Offered fall and spring
Grading System
A - F (Traditional)
Student learning Outcomes
Students will learn general standards and strategies of mathematical proof and apply these to statements about sequences and the real numbers.
Students will learn and be able to explain elements of the theory that provides the foundation of calculus, including precise definitions relating to limits and continuity, and fundamental results such as the intermediate and mean value theorems.
Students will prove rigorously that various kinds of sequences converge, and relate convergence ideas to the behavior of continuous functions.
Students will develop skills in proving and analyzing inequalities, culminating in the error bounds provided by Taylor’s theorem.
Topical Outline
Basics on order, and on inequalities including the triangle inequality
Suprema, infima, and the completeness axiom for the real numbers
Definitions, examples, and theorems related to the convergence of sequences
Subsequences and Cauchy sequences
Series and tests for their convergence
Functional limits and continuity
Proofs of the maximum and intermediate value theorems
Differentiation and the mean value theorem
Taylor polynomials and bounds on the Taylor remainder
