The Infinite Series Module
A Set of Open Resources on Infinite Series
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About the ISM
For Instructors
Technical Requirements
Copyright
Development of the ISM
Units
Unit 1: Sequences and Series
Infinite Sequences
Introduction to Infinite Sequences
Convergence of Infinite Sequences
Convergence of Infinite Sequences Example
Relationship to Limits of Functions
Limit Laws for Infinite Sequences
Limit Laws Example
Relationship to Sequences of Absolute Values
Example Relating Sequences of Absolute Values
Squeeze Theorem for Sequences
Squeeze Theorem Example
Example: {r
n
}
Videos on Infinite Sequences
Final Thoughts on Infinite Sequences
Sigma Notation
Introduction to Sigma Notation
Sigma Notation Terminology
Sigma Notation Properties
Changing Summation Limits
Example: Changing the Summation Limits
Videos on Sigma Notation
Final Thoughts on Sigma Notation
Infinite Series
Introduction to Infinite Series
Convergence of Infinite Series
The Geometric Series
Geometric Series Example
Converting an Infinite Decimal Expansion to a Rational Number
Finding the Sum of an Infinite Series
A Geometric Series Problem with Shifting Indicies
Koch Snowflake Example
Videos on the Introduction to Infinite Series
Final Thoughts on Infinite Series
Properties of Convergent Series
The Properties of Convergent Series
Example: Properties of Convergent Series
The Telescoping and Harmonic Series
Introduction: Telescoping and Harmonic Series
The Harmonic Series
The Telescoping Series
Videos on Telescoping and Harmonic Series
Final Notes on Harmonic and Telescoping Series
Unit 2: Convergence Tests
The Divergence Test
Introduction to the Divergence Test
A Useful Theorem
The Divergence Test
A Divergence Test Flowchart
Simple Divergence Test Example
Divergence Test With Square Roots
Divergence Test with arctan
Video Examples for the Divergence Test
Final Thoughts on the Divergence Test
The Integral Test
A Motivating Problem for The Integral Test
A Second Motivating Problem for The Integral Test
Theorem: The Integral Test
An Integral Test Flowchart
Integral Test Example
Example: Integral Test with a Logarithm
The p-series
Videos on the Integral Test
Final Thoughts on The Integral Test
The Alternating Series Test
A Motivating Problem for the Alternating Series Test
The Alternating Series Test
An Alternating Series Test Concept Map
Return to Our Motivating Example
Alternating Series Test Videos
The Ratio Test
Introduction to the Ratio Test
The Ratio Test
The Ratio Test Flowchart
A Simple Ratio Test Example
Ratio Test Example with an Exponent
Videos on The Ratio Test
Final Thoughts on the Ratio Test
Review of Convergence Tests
Strategies for Convergence Testing
Example Comparing Two Infinite Series
Convergence Test Videos
Unit 3: Power Series
Power Series
A Motivating Problem for Power Series
The Power Series
Power Series Example
Power Series Convergence
Power Series Convergence Example
Videos on Power Series
Taylor Series
A Motivating Problem
The Taylor Series
Maclaurin Expansion of e
x
Maclaurin Expansion of sin(x)
The Maclaurin Expansion of cos(x)
List of Maclaurin Expansions
Videos on Taylor Series
Appendices
Recommended Textbooks and Course Notes
The Contrapositive and the Divergence Test
The Definition of the Contrapositive
Contrapositive Examples
Contrapositive Example with Sets
Back to The Divergence Test
Proof of the Ratio Test
Absolute Convergence
Absolute Convergence Implies Convergence
Proof of the Ratio Test
List of Videos in the ISM
Download
Quizzes
Quiz 1
Quiz 2
Quiz 3
Contact
Video Examples for the Divergence Test
[previous] [next]
Patrick JMT: Showing a Series Diverges Using its Partial Sums
In this video, Patrick JMT explores an example of what can be done when the divergence test fails. In this case, using partial sums can tell us whether a series diverges.
More Patrick JMT videos available
here
.
Patrick JMT: Geometric Series and the Test for Divergence (Part 1)
In this video, Patrick JMT discusses two geometric series examples. The example at the very end is completed in the next video.
More Patrick JMT videos available
here
.
Patrick JMT: Geometric Series and the Test for Divergence (Part 2)
In this video, Patrick JMT finishes the example he started in the previous video.
More Patrick JMT videos available
here
.
[previous] [next]
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Site Map
Home
About the ISM
Units
Unit 1: Sequences and Series
Unit 2: Convergence Tests
The Divergence Test
Introduction to the Divergence Test
A Useful Theorem
The Divergence Test
A Divergence Test Flowchart
Simple Divergence Test Example
Divergence Test With Square Roots
Divergence Test with arctan
Video Examples for the Divergence Test
Final Thoughts on the Divergence Test
The Integral Test
The Alternating Series Test
The Ratio Test
Review of Convergence Tests
Unit 3: Power Series
Appendices
Download
Quizzes
Contact
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