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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

Introduction to Series

An series is an infinite sum, which we think of as the sum of the terms of a sequence, $a_1 + a_2 + a_3 + \ldots.$  We write a series using summation notation as $$\sum_{n=1}^\infty a_n=a_1 + a_2 + a_3 + \cdots.$$DO:  Convince yourself that  $\displaystyle\sum_{i=1}^\infty a_i=\sum_{k=1}^\infty a_k=\sum_{n=1}^\infty a_n=a_1+a_2+a_3+\cdots $.  The $i,k$, and $n$ are just indices.
a sequence is a list of infinitely many terms $a_1,a_2,a_2\ldots$, but 
a series
is a sum of infinitely many terms $a_1 + a_2 + a_3 + \cdots$.

Addition is something we do to finitely many numbers, not to infinitely many - how can we add infinitely many numbers?  Consider a series $\displaystyle\sum_{i=1}^\infty a_i=a_1 + a_2 + a_3 + \cdots.$  Instead of worrying about adding infinitely many $a_i$, we take (finite) partial sums $s_n$, where $s_n$ is the sum of the first $n$ terms of the series.$$\begin{eqnarray*}s_1&=&a_1\\s_2&=&a_1+a_2\\s_3&=&a_1+a_2+a_3\\&\vdots&\\s_n&=&a_1+a_2+\cdots+a_{n-1}+a_n\\   \end{eqnarray*}$$A partial sum makes perfect sense as a sum of finitely many numbers.  We can, of course, write each of these partial sums in summation notation; for example  $$ s_2=\sum_{i=1}^2a_i, \qquad s_5=\sum_{i=1}^5 a_i,\qquad s_n=\sum_{i=1}^n a_i, \quad\text{ etc.}$$Notice that as $n$ gets larger and larger, $\displaystyle s_n=\sum_{i=1}^n=a_1+a_2+\cdots+a_n$ gets closer and closer to the infinite sum $\displaystyle \sum_{i=1}^\infty a_i$. 

Since we cannot add infinitely many things, we make sense of the series by taking a limit.  We define the series as the limit of the partial sums as $n$ goes to infinity.

Let $\displaystyle\displaystyle\sum_{i=1}^\infty a_i$ be a series, and let $\displaystyle s_n = \sum_{i=1}^n a_i$ be its  $n^{th}$ partial sum
We define $\displaystyle\sum_{i=1}^\infty a_i = \lim_{n \to \infty} s_n.$
If this limit exists and is finite and equal to $s$, we say the series is convergent and that $\displaystyle\sum_{i=1}^\infty a_i=s$.  Otherwise, we say the series is divergent.
Often, we can tell that a series converges, but cannot tell what value it converges to.  For most of these modules on series, we will be sharpening our notions of convergence of series.

Warning:  It may be confusing, but we often say $\displaystyle \sum_{n=1}^\infty a_n = \lim_{n \to \infty} s_n$, where the indicies $n$ on the left and on the right are not the same.  It is normal for us to define a series $\sum a_n$ in terms of $n$ and partial sums $s_n$ in terms of $n$ as well, but the $n$'s are used differently.

In the video, much of this will be discussed.

Note: There is an error at the 8:05 mark of the video. The voice-over is correct, but what is written is a little different. The correct values are $s_4=-2$, $s_5=3$ and $s_6=-3$. This doesn't affect the point being made about divergence.