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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

Strategy to Test Series and a Review of Tests

Notation:  In this section, we will often use the following series notations:
$\displaystyle\sum_{n}^\infty a_n=\sum_n a_n=\sum a_n$. 
&\text{All of these notations indicate that the index is $n$,}\\
&\text{but we aren't declaring where $n$ begins ($n=0$ or $n=1$ or $n=5$ etc.).}\\

As with techniques of integration, it is important to recognize the form of a series in order to decide your next steps.  Although there are no hard-and-fast rules, running down the following steps (in order) may be helpful.

Strategy to test series

  1. If you see that the terms $a_n$ do not go to zero, you know the series diverges by the Divergence Test.

  2. If a series is a $p$-series, with terms $\frac{1}{n^p}$, we know it converges if $p>1$ and diverges otherwise.

  3. If a series is a geometric series, with terms $ar^n$, we know it converges if $|r|<1$ and diverges otherwise.  In addition, if it converges and the series starts with $n=0$ we know its value is $\frac{a}{1-r}$.  (If it starts with another value of $n$, some work must be done to determine its value.)

  4. If a series is similar to a $p$-series or a geometric series, you should consider a Comparison Test or a Limit Comparison Test.  These test only work with positive term series, but if your series has both positive and negative terms you can test $\sum|a_n|$ for absolute convergence.

  5. If the series has alternating signs, the Alternating Series Test is helpful; in particular, in a previous step you have already determined that your terms go to zero.  However, the AST will not indicate whether a series converges absolutely or conditionally - determining this will require other tests.

  6. If your terms contain factorials, or factorials and $n^{th}$ powers, the Ratio Test might be helpful.  This test does not care if your terms are negative, and may determine absolute convergence of the series.  However, this test will fail for $p$-series and all rational functions of $n$, so don't try the Ratio Test on these.

  7. If your terms contain $n^{th}$ powers, the Root Test may be helpful.  (If you have a geometric series, you will already know it before coming to this step.)  This test does not care if your terms are negative, and may determine Absolute Convergence of the series.

  8. If your terms are positive and decreasing, and easily integrated (when viewed as $f(x)$ where $f(n)=a_n$), the Integral Test may be helpful.

A review of all series tests

Consider the series $\displaystyle\sum_{n}^\infty a_n$.

Divergence Test: If $\displaystyle\lim_{n \to \infty} a_n \ne 0$, then $\displaystyle\sum_n a_n$ diverges.

Integral Test: If $a_n = f(n)$, where $f(x)$ is a non-negative non-increasing function, then 
$\displaystyle\sum_{n}^\infty a_n$ converges if and only if the integral $\displaystyle\int_1^\infty f(x) \,dx$ converges.

Comparison Test: This applies only to positive-term series.
If $a_n \le b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges.

If $b_n \le a_n$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

Limit comparison Test: If $\sum a_n$ and $\sum b_n$ are positive-term series, and
$\displaystyle\lim_{n \to \infty} \frac{a_n}{b_n} = L$, with $0<L<\infty$, then either
$\sum a_n$ and $\sum b_n$ both converge or both diverge.

Alternating Series Test:  When our series is alternating, so that $\displaystyle\sum_n^\infty a_n=\sum_n^\infty(-1)^nb_n$, if
$b_n>0$, $\quad b_{n+1} \le b_n,\quad$ and $\quad\displaystyle\lim_{n \to \infty}b_n = 0$, then
$\sum (-1)^{n+1} b_n$ converges.

Ratio Test: Let $L= \displaystyle{\lim_{n\to\infty} \frac{|a_{n+1}|}{|a_n|}}$.
If $L < 1$, then $\sum a_n$ converges absolutely.
If $L > 1$, or the limit goes to $\infty$, then $\sum a_n$ diverges.
If $L=1$ or if $L$ does not exist, then the test fails, and we know nothing.

Root Test: Let $L = \displaystyle\lim_{n \to \infty}\sqrt[n]{|a_n|}$.
If $L<1$, then $\sum a_n$ converges absolutely.
If $L>1$,  or the limit goes to infinity, then $\sum a_n$ diverges.
If $L=1$, or if $L$ does not exist, then the test fails, and we know nothing.