Explore an in-depth 8-hour course on sequences and series in calculus. Begin with an introduction to sequences, learning how to list terms, find formulas, and understand limits. Dive into limit properties, bounded sequences, and monotone sequences with practical examples. Progress to series, covering divergent series, geometric series, p-series, and telescoping series. Master integral and comparison tests, including limit comparison. Study the ratio and root tests for series convergence. Examine alternating series, absolute convergence, and approximation theorems. Delve into Taylor polynomials, understanding their coefficients, definitions, and error estimates. Investigate power series, exploring their radius of convergence, derivatives, and integrals. Conclude with an extensive look at Taylor series, including Maclaurin series, natural logarithms, binomial series, and techniques for manipulating known series. Apply your knowledge to solve initial value problems and integrate series.
Overview
Syllabus
Sequences and Series: Sequences - 01. Intro.
Sequences and Series: Sequences - 02. Example: Listing Terms.
Sequences and Series: Sequences - 03. Example: Finding Formulas.
Sequences and Series: Sequences - 04. Limits.
Sequences and Series: Sequences - 05. Example: Limits.
Sequences and Series: Sequences - 06. Limit Properties.
Sequences and Series: Sequences - 07. Example: Limit Properties.
Sequences and Series: Sequences - 08. Bounded Sequences.
Sequences and Series: Sequences - 09. Convergent Sequences are Bounded.
Sequences and Series: Sequences - 10. Monotone Sequences.
Sequences and Series: Sequences - 11. Example: Monotone sequences.
Sequences and Series: Sequences - 12. Example: Limit of a bounded monotone sequence.
Sequences and Series: Introduction to Series - 01. Intro.
Sequences and Series: Introduction to Series - 02. Example: Divergent series.
Sequences and Series: Introduction to Series - 03. Geometric series.
Sequences and Series: Introduction to Series - 04. Example: Geometric series.
Sequences and Series: Introduction to Series - 05. p-Series.
Sequences and Series: Introduction to Series - 06. Telescoping series.
Sequences and Series: Introduction to Series - 07. Example: Telescoping series.
Sequences and Series: Introduction to Series - 08. Example: Evaluate using properties.
Sequences and Series: Integral and Comparison Tests - 01. Integral Test.
Sequences and Series: Integral and Comparison Tests - 02. Example - Integral Test 1.
Sequences and Series: Integral and Comparison Tests - 03. Example - proving the p-Test.
Sequences and Series: Integral and Comparison Tests - 04. Comparison Test.
Sequences and Series: Integral and Comparison Tests - 05. Example - Comparison.
Sequences and Series: Integral and Comparison Tests - 06. Example - Comparison Test 2.
Sequences and Series: Integral and Comparison Tests - 07. Example - Motivating Limit Comparison.
Sequences and Series: Integral and Comparison Tests - 08. Limit Comparison.
Sequences and Series: Integral and Comparison Tests - 09. Example: Limit Comparison 1.
Sequences and Series: Integral and Comparison Tests - 10. Example - Limit Comparison 2.
Sequences and Series: Integral and Comparison Tests - 11. Example: Limit Comparison 3.
Sequences and Series: Ratio and Root Tests - 01. Ratio Test.
Sequences and Series: Ratio and Root Tests - 02. Example.
Sequences and Series: Ratio and Root Tests - 03. Example 2.
Sequences and Series: Ratio and Root Tests - 04. Root Test.
Sequences and Series: Ratio and Root Tests - 05. Examples.
Sequences and Series: Alternating Series - 01. Definition and Theorem.
Sequences and Series: Alternating Series - 02. Examples.
Sequences and Series: Alternating Series - 03. Approximation Theorem.
Sequences and Series: Alternating Series - 04. Example.
Sequences and Series: Alternating Series - 05. Absolute Convergence.
Sequences and Series: Alternating Series - 06. Examples.
Derivative Applications: Taylor polynomials - 01. Intro.
Derivative Applications: Taylor polynomials - 02. Determining Coefficients.
Derivative Applications: Taylor polynomials - 03. Definition.
Derivative Applications: Taylor polynomials - 04. Example.
Derivative Applications: Taylor polynomials - 05. Example.
Derivative Applications: Taylor polynomials - 06. Example.
Derivative Applications: Taylor polynomials - 07. Taylor's Theorem.
Derivative Applications: Taylor polynomials - 08. Error Estimate Example.
Derivative Applications: Taylor polynomials - 09. Example.
Sequences and Series: Power Series - 01. Introduction.
Sequences and Series: Power Series - 02. Examples.
Sequences and Series: Power Series - 03. Radius of Convergence.
Sequences and Series: Power Series - 05. Derivatives and Integrals.
Sequences and Series: Power Series - 04. Example: Radius of Convergence.
Sequences and Series: Power Series - 06. Example.
Sequences and Series: Power Series - 07. Example: Exponential function.
Sequences and Series: Taylor Series - 01. Introduction.
Sequences and Series: Taylor Series - 02. Example: Maclaurin series for cosine.
Sequences and Series: Taylor Series - 03: Example: natural log.
Sequences and Series: Taylor Series - 04. Equality of series and function.
Sequences and Series: Taylor Series - 05. Binomial Series - Introduction.
Sequences and Series: Taylor Series - 06. Binomial Series.
Sequences and Series: Taylor Series - 07. Binomial Series - Convergence and Examples.
Sequences and Series: Taylor Series - 08. Manipulating known series - Intro.
Sequences and Series: Taylor Series - 09. Manipulating known series - Product.
Sequences and Series: Taylor Series - 10. Manipulating known series - Composition.
Sequences and Series: Taylor Series - 11. Integration.
Sequences and Series: Taylor Series - 12. Solving an Initial Value Problem.
Taught by
Sean Fitzpatrick
