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Last updated:

November 4, 2022

Duration:

Unlimited Duration

FREE

This course includes:

Unlimited Duration

Badge on Completion

Certificate of completion

Unlimited Duration

Description

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

Course Curriculum

  • 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

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Massachusetts Institute of Technology