5. Exploring the Infinite

Course Info

Instructor

Prof. David Jerison

Departments

Mathematics

As Taught In

Fall 2010

Level

Undergraduate

Topics

Mathematics
- Calculus
- Differential Equations

Learning Resource Types

Exams with Solutions

Lecture Notes

Lecture Videos

Problem Sets with Solutions

Simulations

Recitation Videos

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Introduction

The key concept upon which Calculus is built is the notion of “limits.” This unit covers topics related to limits, infinitely many, or infinitely small values.

Part A: L’Hospital’s Rule and Improper Integrals

Part B: Taylor Series

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L’Hospital’s rule describes how we can use the derivative to calculate certain limits. The second half of this part of the course discusses how to calculate the area under a graph when that area is unbounded.

» Session 87: L’Hospital’s Rule
» Session 88: Examples of L’Hospital’s Rule
» Session 89: L’Hospital’s Rule and Rates of Growth
» Session 90: Advanced Examples of L’Hospital’s Rule
» Session 91: Improper Integrals
» Session 92: Integral Comparison
» Session 93: Indefinite Integrals and Singularities

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We can use limits to formally define sums of infinitely many values. Taylor Series are infinite sums defined in terms of polynomials, and provide an alternate means of describing standard functions.

» Session 94: Infinite Series
» Session 95: Series Comparison
» Session 96: Stacking Blocks
» Session 97: Power Series
» Session 98: Taylor’s Series
» Session 99: Taylor’s Series, Continued
» Session 100: Operations on Power Series
» Session 101: Conclusion