This course covers the learning outcomes and goals of understanding sequences and series of real numbers and functions. Students will learn to manipulate convergent sequences, explore bounds, study monotone sequences, and understand the Bolzano-Weierstrass Theorem. The course also delves into sequences of functions, uniform convergence, Taylor series, and various tests for convergence of series. The teaching method involves theoretical explanations, proofs, and examples. This course is intended for learners interested in advanced mathematics, particularly those studying real analysis or calculus.
Sequences and Series of Real Numbers and Functions
Overview
Syllabus
Adding, Multiplying, and Dividing Convergent Sequences of Real Numbers.
Bounds on a Convergent Sequence of Real Numbers.
Sandwiching Sequences of Real Numbers.
Monotone Sequences of Real Numbers.
Subsequences of Real Numbers.
Bolzano Weierstrass Theorem.
Cauchy Sequence of Real Numbers.
Limit Supremum and Limit Infimum of a Sequence of Real Numbers.
Sequences of Functions - Pointwise and Uniform Convergence.
Uniform Convergence of Continuous Functions.
Uniform Convergence of Riemann Integrable Functions.
Taylor Series - Integral Form of the Remainer.
Geometric Series.
Prove that the Sequence {(1+1/n)^n} is Convergent using Sequence Theory.
Uniform Convergence of Bounded Functions.
Uniform Convergence of Differentiable Functions.
Series of Real Numbers.
Comparison Test for Convergence of a Series.
Cauchy–Schwarz Inequality.
Weierstrass M Test.
Ratio Test for Convergence of a Series.
Root Test for Convergence of a Series.
The Ratio Test Implies The Root Test.
Integral Test for Convergence of a Series.
Conditional and Absolute Convergence.
Cauchy Product.
Mertens' Theorem - Product of 2 Infinite Series.
Limit Supremum and Limit Infimum of Sets (part 1 of 2).
Limit Supremum and Limit Infimum of Sets (part 2 of 2).
Proof of Stirling's Formula for an approximation of n!.
Sequences that Limit to Zero, But the Series Diverges. #TeamSeas.
2 Examples with limsup and liminf.
Taught by
statisticsmatt
