Math 312 : Concepts of Real Analysis

Fall, 2014

http://www.math.psu.edu/li/teaching.html


PROFESSOR: Dr. Jenny X. Li
Office: 220 McAllist Building
Phone: 863-9106
office hours: Tue. 11:30-12:30, Friday 11:30-12:30 or by appointment
E-mail: li@math.psu.edu
COURSE: An introduction to rigorous mathematical proofs which involving properties of real numbers, sequences, continuity, series of function, differentiation and integration.
TEXTBOOK: Elementary Analysis: The Theory of Calculus by Kenneth A. Ross 2th edition., Springer
REFERENCE:Introduction to Analysis by Arthur Mattuck, latest edition
PREREQUISITES: Math 141
GRADE:
  1. Homework and Quizzes (25%): Every week.
  2. Midterm (35%): Oct.15 at regular lecture time.
  3. Final Exam (40%): Time and place will be announced.
POLICIES:
  1. NO late homework will be accepted, but the lowest grade of homework will be dropped. Also there will be NO makeup midterm.
  2. Quizzes will occur whenever it is necessary. They will be announced in advance, also No makeup quizzes.
  3. Bonus questions will be given throughout the semester which will be added to the final grade.
  4. NO books, notes, calculators or other electronic devices are permitted in the quizzes, midterm or final exam.
TENTATIVE OUTLINE:
  • PART I Introduction

  • Review
  • The sets of Naturally and Rational Numbers
  • The set of Real Numbers
  • The Completeness Axiom

  • PART II Sequences

  • Limits of Sequence
  • Monotone Sequences and Cauchy Sequences
  • Lim sup's and lim inf's
  • Alternating Series and Integral Tests

  • PART III Continuity

  • Continuous Functions
  • Properties of Continuous Functions
  • Uniform Continuity
  • Limits of Functions
  • Topological Concepts in Metric Spaces
  • More on Metric Space: Continuity and Connectedness

  • PART IV Sequences and Series of Functions

  • Uniform Convergence
  • Differentiation and Integration of Power Series
  • Weierstrass's Approximation Theorem

  • PART V Differentiation

  • Basic Properties of the Derivative
  • The Mean Value Theorem
  • L'Hospital's Rule
  • Taylor's Theorem