18.100A Real Analysis (Fall 2020, MIT OCW). Instructor: Dr. Casey Rodriguez. This course covers the fundamentals of mathematical analysis
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September 21, 2023
English
English [CC]
Description
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. (from ocw.mit.edu)
Course Curriculum
- Lecture 01 – Sets, Set Operations and Mathematical Induction Unlimited
- Lecture 02 – Cantor’s Theory of Cardinality (Size) Unlimited
- Lecture 03 – Cantor’s Remarkable Theorem and the Rationals’ Lack of the Least … Unlimited
- Lecture 04 – The Characterization of the Real Numbers Unlimited
- Lecture 05 – The Archimedean Property, Density of the Rotations, and Absolute Value Unlimited
- Lecture 06 – The Uncountability of the Real Numbers Unlimited
- Lecture 07 – Convergent Sequences of Real Numbers Unlimited
- Lecture 08 – The Squeeze Theorem and Operations Involving Convergent Sequences Unlimited
- Lecture 09 – Limsup, Liminf, and the Bolzano-Weierstrass Theorem Unlimited
- Lecture 10 – The Completeness of the Real Numbers and Basic Properties of Infinite Series Unlimited
- Lecture 11 – Absolute Convergence and the Comparison Test for Series Unlimited
- Lecture 12 – The Ratio, Root, and Alternating Series Tests Unlimited
- Lecture 14 – Limits of Functions in terms of Sequences and Continuity Unlimited
- Lecture 15 – The Continuity of Sine and Cosine and the Many Discontinuities of … Unlimited
- Lecture 16 – The Min/Max Theorem and Bolzano’s Intermediate Value Theorem Unlimited
- Lecture 17 – Uniform Continuity and the Definition of the Derivative Unlimited
- Lecture 18 – Weierstrass’ Example of a Continuous and Nowhere Differentiable Function Unlimited
- Lecture 19 – Differentiation Rules, Rolle’s Theorem, and the Mean Value Theorem Unlimited
- Lecture 20 – Taylor’s Theorem and the Definition of Riemann Sums Unlimited
- Lecture 21 – The Riemann Integral of a Continuous Function Unlimited
- Lecture 22 – Fundamental Theorem of Calculus, Integration by Parts, and … Unlimited
- Lecture 23 – Pointwise and Uniform Convergence of Sequences of Functions Unlimited
- Lecture 24 – Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits Unlimited
- Lecture 25 – Power Series and the Weierstrass Approximation Theorem Unlimited
About the instructor
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