18.100A Real Analysis (Fall 2020, MIT OCW): Lecture 24 - Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits

18.100A Real Analysis

18.100A Real Analysis (Fall 2020, MIT OCW). Instructor: Dr. Casey Rodriguez. 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. (from ocw.mit.edu)

Lecture 24 - Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits

Instructor: Dr. Casey Rodriguez. We prove the powerful Weierstrass M-test, and within our setting of sequences of continuous or differentiable functions, we address a fundamental question of analysis: when can we interchange two limits?

Go to the Course Home or watch other lectures:

Lecture 01 - Sets, Set Operations and Mathematical Induction

Lecture 02 - Cantor's Theory of Cardinality (Size)

Lecture 03 - Cantor's Remarkable Theorem and the Rationals' Lack of the Least Upper Bound Property

Lecture 04 - The Characterization of the Real Numbers

Lecture 05 - The Archimedean Property, Density of the Rotations, and Absolute Value

Lecture 06 - The Uncountability of the Real Numbers

Lecture 07 - Convergent Sequences of Real Numbers

Lecture 08 - The Squeeze Theorem and Operations Involving Convergent Sequences

Lecture 09 - 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