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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 for Indefinite Integrals
Examples to Try
Revised Table of Integrals
Substitution for Definite Integrals

Area Between Curves

Computation Using Integration
To Compute a Bulk Quantity
The Area Between Two Curves
Horizontal Slicing


Slicing and Dicing Solids
Solids of Revolution 1: Disks
Solids of Revolution 2: Washers
More Practice

Integration by Parts

Integration by Parts
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
Completing the Square

Partial Fractions

Linear Factors
Irreducible Quadratic Factors
Improper Rational Functions and Long Division

Strategies of Integration

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

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

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

The Ratio and Root Tests

The Ratio Test
The Root Test

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

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

Test for Divergence and Other Theorems

The following test is very valuable if the terms of your series do not converge to zero. 

Test for divergence

  • If $\displaystyle\sum_{n=1}^\infty a_n$ converges, then it must be true that $\displaystyle\lim_{n \to \infty} a_n = 0$.
  • Equivalently, if $\displaystyle\lim_{n \to \infty} a_n$ is not zero, then $\displaystyle\sum_{n=1}^\infty a_n$ diverges.

The converse is not true.  Just because $\lim a_n=0$ it is not necessarily true that $\sum a_n$ converges.  It is necessary that your terms go to zero in order to have a convergent series, but this is not enough to ensure convergence.  In other words, this test for divergence can only be used to test for divergence; it does not help us determine convergence. 

For example, the divergence test tells us that $\sum 2^n$ diverges, since our terms $2^n$ certainly do not converge to 0.  The test does not help with the series $\sum\tfrac 1 n$, since the terms go to zero.  It turns out that this series, called the harmonic series, does not converge.  Similarly, we cannot use the test with $\sum\tfrac 1{n^2}$ since its terms go to 0, but it turns out that this series does converge.

Examples:  Try to use the test for divergence for $\displaystyle\sum_{n=0}^\infty\frac{n^2}{2n^2-5}$ and $\displaystyle\sum_{n=1}^\infty\ln\left(\frac{n}{n-2}\right)$.  DO before looking at the solutions.

Solutions:  $\displaystyle\lim_{n\to\infty}\frac{n^2}{2n^2-5}=\frac{1}{2}\not = 0$, so the series diverges by the test for divergence.  $\displaystyle\lim_{n\to\infty}\ln\left(\frac{n}{n-2}\right)=\ln(1)=0$.  Since our terms go to zero, the divergence test does not help, and we don't know if the series converges or diverges without doing more work.

Because the definition of a convergent series is a limit, we have the following theorems about convergent series
$\displaystyle\sum_{n=1}^\infty a_n = L$ and $\displaystyle\sum_{n=1}^\infty b_n = M$.  Warning:  If either series is divergent, we do not have these facts.

$If \displaystyle\sum_{n=1}^\infty a_n = L$ and $\displaystyle\sum_{n=1}^\infty b_n = M,$
$\displaystyle\sum_{n=1}^\infty \left(a_n+b_n\right) = L+M$, $\displaystyle\sum_{n=1}^\infty \left(a_n-b_n\right) = L-M$, and
$\displaystyle\sum_{n=1}^\infty \left(c\, a_n\right) = cL$. 
There is no similar information for $\displaystyle\sum_{n=1}^\infty\left( a_nb_n\right)$ or $\displaystyle\sum_{n=1}^\infty \left(\frac{a_n}{b_n}\right)$.

This video justifies these theorems.