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Massachusetts Institute of Technology

Real Analysis

Massachusetts Institute of Technology via MIT OpenCourseWare

Overview

This course covers the fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts through a study of real numbers, and teaches an understanding and construction of proofs.

Syllabus

  • Lecture 1: Sets, Set Operations and Mathematical Induction
  • Lecture 2: Cantor's Theory of Cardinality (Size)
  • Lecture 3: Cantor's Remarkable Theorem and the Rationals' Lack of the Least Upper Bound Property
  • Lecture 4: The Characterization of the Real Numbers
  • Lecture 5: The Archimedian Property, Density of the Rationals, and Absolute Value
  • Lecture 6: The Uncountabality of the Real Numbers
  • Lecture 7: Convergent Sequences of Real Numbers
  • Lecture 8: The Squeeze Theorem and Operations Involving Convergent Sequences
  • Lecture 9: Limsup, Liminf, and the Bolzano-Weierstrass Theorem
  • Lecture 10: The Completeness of the Real Numbers and Basic Properties of Infinite Series
  • Lecture 11: Absolute Convergence and the Comparison Test for Series
  • Lecture 12: The Ratio, Root, and Alternating Series Tests
  • Lecture 13: Limits of Functions
  • Lecture 14: Limits of Functions in Terms of Sequences and Continuity
  • Lecture 15: The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet's Function
  • Lecture 16: The Min/Max Theorem and Bolzano's Intermediate Value Theorem
  • Lecture 17: Uniform Continuity and the Definition of the Derivative
  • Lecture 18: Weierstrass's Example of a Continuous and Nowhere Differentiable Function
  • Lecture 19: Differentiation Rules, Rolle's Theorem, and the Mean Value Theorem
  • Lecture 20: Taylor's Theorem and the Definition of Riemann Sums
  • Lecture 21: The Riemann Integral of a Continuous Function
  • Lecture 22: Fundamental Theorem of Calculus, Integration by Parts, and Change of Variable Formula
  • Lecture 23: Pointwise and Uniform Convergence of Sequences of Functions
  • Lecture 24: Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits
  • Lecture 25: Power Series and the Weierstrass Approximation Theorem

Taught by

Dr. Casey Rodriguez

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