MIT 18.100A Real Analysis, Fall 2020 (English)

   Monday, December 02, 2024   

Starting a new Lecture Notes Series on MIT 18.100A Real Analysis, Fall 2020

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MIT 18.100A Real Analysis, Fall 2020 By Lecture Notes together!

Lecture 1: Lecture 1: Sets, Set Operations and Mathematical Induction

Lecture 2: Lecture 2: Cantor's Theory of Cardinality (Size)

Lecture 3: Lecture 3: Cantor's Remarkable Theorem and the Rationals' Lack of the Least Upper Bound Property

Lecture 4: Lecture 4: The Characterization of the Real Numbers

Lecture 5: Lecture 5: The Archimedian Property, Density of the Rationals, and Absolute Value

Lecture 6: Lecture 6: The Uncountabality of the Real Numbers

Lecture 7: Lecture 7: Convergent Sequences of Real Numbers

Lecture 8: Lecture 8: The Squeeze Theorem and Operations Involving Convergent Sequences

Lecture 9: Lecture 9: Limsup, Liminf, and the Bolzano-Weierstrass Theorem

Lecture 10: Lecture 10: The Completeness of the Real Numbers and Basic Properties of Infinite Series

Lecture 11: Lecture 11: Absolute Convergence and the Comparison Test for Series

Lecture 12: Lecture 12: The Ratio, Root, and Alternating Series Tests

Lecture 13: Lecture 13: Limits of Functions

Lecture 14: Lecture 14: Limits of Functions in Terms of Sequences and Continuity

Lecture 15: Lecture 15: The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet's Function

Lecture 16: Lecture 16: The Min/Max Theorem and Bolzano's Intermediate Value Theorem

Lecture 17: Lecture 17: Uniform Continuity and the Definition of the Derivative

Lecture 18: Lecture 18: Weierstrass's Example of a Continuous and Nowhere Differentiable Function

Lecture 19: Lecture 19: Differentiation Rules, Rolle's Theorem, and the Mean Value Theorem

Lecture 20: Lecture 20: Taylor's Theorem and the Definition of Riemann Sums

Lecture 21: Lecture 21: The Riemann Integral of a Continuous Function

Lecture 22: Lecture 22: Fundamental Theorem of Calculus, Integration by Parts, and Change of Variable Formula

Lecture 23: Lecture 23: Pointwise and Uniform Convergence of Sequences of Functions

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

Lecture 25: Lecture 25: Power Series and the Weierstrass Approximation Theorem