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Lecture 17: Taylor Polynomials; Remainder Term; Riemann Integrals
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Classroom Contents
Real Analysis - Spring 2025
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- 1 Lecture 1: Introduction to Real Numbers
- 2 Lecture 2: Introduction to Real Numbers (cont.)
- 3 Lecture 3: How to Write a Proof; Archimedean Property
- 4 Lecture 4: Sequences; Convergence
- 5 Lecture 5: Monotone Convergence Theorem
- 6 Lecture 6: Cauchy Convergence Theorem
- 7 Lecture 7: Bolzano–Weierstrass Theorem; Cauchy Sequences; Series
- 8 Lecture 8: Convergence Tests for Series; Power Series
- 9 Lecture 9: Limsup and Liminf; Power Series; Continuous Functions; Exponential Function
- 10 Lecture 10: Continuous Functions; Exponential Function (cont.)
- 11 Lecture 11: Extreme and Intermediate Value Theorem; Metric Spaces
- 12 Review for 18.100B Real Analysis Midterm
- 13 Lecture 12: Convergence in Metric Spaces; Operations on Sets
- 14 Lecture 13: Open and Closed Sets; Coverings; Compactness
- 15 Lecture 14: Sequential Compactness; Bolzano–Weierstrass Theorem in a Metric Space
- 16 Lecture 15: Derivatives; Laws for Differentiation
- 17 Lecture 16: Rolle’s Theorem; Mean Theorem; L’Hôpital’s Rule; Taylor Expansion
- 18 Lecture 17: Taylor Polynomials; Remainder Term; Riemann Integrals
- 19 Lecture 18: Integrable Functions
- 20 Lecture 19: Fundamental Theorem of Calculus
- 21 Lecture 20: Pointwise Convergence; Uniform Convergence
- 22 Lecture 21: Integrals and Derivatives under Uniform Convergence
- 23 Lecture 22: Differentiating and Integrating Power Series; Ordinary Differential Equations (ODEs)
- 24 Lecture 23: Existence & Uniqueness for ODEs: Picard–Lindelöf Theorem
- 25 Review for the 18.100B Real Analysis Final Exam
