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The Fundamental Theorem of Calculus
Three Different Concepts
The Fundamental Theorem of Calculus (Part 2)
The Fundamental Theorem of Calculus (Part 1)
More FTC 1
The Indefinite Integral and the Net Change
Indefinite Integrals and Anti-derivatives
A Table of Common Anti-derivatives
The Net Change Theorem
The NCT and Public Policy
Substitution
Substitution for Indefinite Integrals
Examples to Try
Revised Table of Integrals
Substitution for Definite Integrals
Examples
Area Between Curves
Computation Using Integration
To Compute a Bulk Quantity
The Area Between Two Curves
Horizontal Slicing
Summary
Volumes
Slicing and Dicing Solids
Solids of Revolution 1: Disks
Solids of Revolution 2: Washers
More Practice
Integration by Parts
Integration by Parts
Examples
Integration by Parts with a definite integral
Going in Circles
Tricks of the Trade
Integrals of Trig Functions
Antiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only
odd powers)
Product of Sines and Cosines (only even powers)
Product of Secants and Tangents
Other Cases
Trig Substitutions
How Trig Substitution Works
Summary of trig substitution options
Examples
Completing the Square
Partial Fractions
Introduction
Linear Factors
Irreducible Quadratic Factors
Improper Rational Functions and Long Division
Summary
Strategies of Integration
Substitution
Integration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions
Improper Integrals
Type 1 - Improper Integrals with Infinite Intervals of
Integration
Type 2 - Improper Integrals with Discontinuous Integrands
Comparison Tests for Convergence
Differential Equations
Introduction
Separable Equations
Mixing and Dilution
Models of Growth
Exponential Growth and Decay
Logistic Growth
Infinite Sequences
Approximate Versus Exact Answers
Examples of Infinite Sequences
Limit Laws for Sequences
Theorems for and Examples of Computing Limits of Sequences
Monotonic Covergence
Infinite Series
Introduction
Geometric Series
Limit Laws for Series
Test for Divergence and Other Theorems
Telescoping Sums and the FTC
Integral Test
Road Map
The Integral Test
Estimates of Value of the Series
Comparison Tests
The Basic Comparison Test
The Limit Comparison Test
Convergence of Series with Negative Terms
Introduction, Alternating Series,and the AS Test
Absolute Convergence
Rearrangements
The Ratio and Root Tests
The Ratio Test
The Root Test
Examples
Strategies for testing Series
Strategy to Test Series and a Review of Tests
Examples, Part 1
Examples, Part 2
Power Series
Radius and Interval of Convergence
Finding the Interval of Convergence
Power Series Centered at $x=a$
Representing Functions as Power Series
Functions as Power Series
Derivatives and Integrals of Power Series
Applications and Examples
Taylor and Maclaurin Series
The Formula for Taylor Series
Taylor Series for Common Functions
Adding, Multiplying, and Dividing Power Series
Miscellaneous Useful Facts
Applications of Taylor Polynomials
Taylor Polynomials
When Functions Are Equal to Their Taylor Series
When a Function Does Not Equal Its Taylor Series
Other Uses of Taylor Polynomials
Partial Derivatives
Visualizing Functions in 3 Dimensions
Definitions and Examples
An Example from DNA
Geometry of Partial Derivatives
Higher Order Derivatives
Differentials and Taylor Expansions
Multiple Integrals
Background
What is a Double Integral?
Volumes as Double Integrals
Iterated Integrals over Rectangles
How To Compute Iterated Integrals
Examples of Iterated Integrals
Cavalieri's Principle
Fubini's Theorem
Summary and an Important Example
Double Integrals over General Regions
Type I and Type II regions
Examples 1-4
Examples 5-7
Order of Integration
Examples, Part 1
DO: Test the following series for
convergence, divergence, conditional convergence and absolute
convergence, when possible. Hint: One way to proceed
with the first two series is by splitting the terms into three
parts. Remember, if $\sum a_n$ and $\sum b_n$ both converge,
then $\sum a_n+\sum
b_n=\sum(a_n+b_n)$ converges. (Similarly for three series.)
There are more examples to work on the next page, which you might
want to work on also before checking any answers.
$\displaystyle\sum_n\left(2^{-n} + \frac{3n+5}{n^3}\right)$;
$\displaystyle\sum_n \left(2^{-n} + \frac{3n+5}{n^2}\right)$;
$\displaystyle\sum_n \left(2^{-n} + \frac{3n+5}{n}\right)$;
$\displaystyle\sum_n \frac{1}{n^2+n}$;
$\displaystyle\sum_n \frac{n+\cos(n)}{n^3+n^2}$;
$\displaystyle\sum_n \frac{1}{n \ln(n)}$ ; and
$\displaystyle\sum_n \frac{1}{n \left(\ln(n)\right)^2}$ .
The video below will discuss the preceding series. We have put
the written results below the video.
VIDEO

$\displaystyle\sum_n\left(2^{-n} + \frac{3n+5}{n^3}\right)$
converges as a sum of three
standard examples ;
$\displaystyle\sum_n \left(2^{-n} + \frac{3n+5}{n^2}\right)$
diverges as a sum of three s tandard examples , two of which
converge and one of which diverges;
$\displaystyle\sum_n \left(2^{-n} + \frac{3n+5}{n}\right)$
diverges by the Divergence Test ;
$\displaystyle\sum_n \frac{1}{n^2+n}$ converges by comparison to
$\displaystyle\sum \frac{1}{n^2}$;
$\displaystyle\sum_n \frac{n+\cos(n)}{n^3+n^2}$ also
converges by comparison to $\displaystyle\sum \frac{1}{n^2}$,
but we need the Limit Comparison Test ;
$\displaystyle\sum_n \frac{1}{n \ln(n)}$ diverges by the Integral Test ; and
$\displaystyle\sum_n \frac{1}{n \left(\ln(n)\right)^2}$
converges by the Integral Test .