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This course covers the fundamentals of mathematical analysis:
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English
English [CC]
FREE
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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
Course content
- Sets, set Operations and Mathematical Induction Unlimited
- Cantor’s Theory of Cardinality (size) Unlimited
- Cantor’s Remarkable Theorem and the Rationals’ Lack of the Least Upper Bound Property Unlimited
- The Characterization of the Real Numbers Unlimited
- The Archimedian Propert Density of the Rationals and Absolute value Unlimited
- The Uncountabality of the Real Numbers Unlimited
- Convergent Sequences of Real Numbers Unlimited
- The Sequeeze Theorem and Operations Involving Convergent Sequences Unlimited
- Limsup, Liminf, and the Bolzano-Weierstrass Theorem Unlimited
- The Completness Of the real Numbers and Basic Properties Of Infinite Series Unlimited
- Abolute Convergence and the Comparison Test for Series Unlimited
- The Ratio, Root, and Alternating Series Tests Unlimited
- Limits of Function Unlimited
- Limits Of Functions In terms Of Sequences and Continuity Unlimited
- The Continuity of Sine and and the many Discontinuities of Dirichlet’s Functions Unlimited
- The Min/Max Theorem and Bolzano’s Intermediate Value Theorem Unlimited
- Uniform Continuity and The Definition of The Derivativeva Unlimited
- Weierstrass’s Example of a Continuous and Nowhere Differnetiable Function Unlimited
- Differentiation Rules, Role’s Theorem, and The Mean Value Theorem Unlimited
- Taylor’s Theorem and the Definition of Riemann Sums Unlimited
- The Riemann Integral of a Continuous Function Unlimited
- Fundamental theorem of Calculus, Integration By Parts, and Change of Variable Formula Unlimited
- Pointwise and Uniform Convergence of Sequences of Function Unlimited
- Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits Unlimited
- Power Series and the Weierstrass Approximation Theorem Unlimited
Instructor
Massachusetts Institute of Technology
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