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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

The Ratio Test

Let $\sum a_n$ be a series.  The Ratio Test involves looking at $$\displaystyle{\lim_{n \to \infty} \frac{\left|a_{n+1}\right|}{\left|a_n\right|}}$$ to see how a series behaves in the long run.  As $n$ goes to infinity, this ratio measures how much smaller the value of $a_{n+1}$ is, as compared to the previous term $a_n$, to see how much the terms are decreasing (in absolute value).  If this limit is greater than 1, then for all values of $n$ past a certain point, the ratio $\frac{\left|a_{n+1}\right|}{\left|a_n\right|}>1$, which would indicate that the series is no longer decreasing.  On the other hand, if this limit is less than 1, the series converges absolutely.

The Ratio Test:  Suppose that $$\displaystyle{\lim_{n\to\infty} \frac{\left|a_{n+1}\right|}{\left|a_n\right|}} = L.$$

  • 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 this test is inconclusive, and we must do more work.  We say the Ratio Test fails if $L=1$

Notice that the Ratio Test considers the ratio of the absolute values of the terms.  As you might expect, the Ratio Test thus gives us information about whether the series $\sum a_n$ converges absolutely

Warning:  There are examples with $L=1$ that converge absolutely, examples that converge conditionally, and examples that diverge.  DO:  Apply the Ratio Test to 1) the absolutely convergent series $\sum\frac{1}{n^2}$, 2) the conditionally convergent series $\sum-\frac{1}{n}$, and 3) the divergent series $\sum\frac{1}{n}$.
1)  As $n\to\infty$, $\displaystyle\frac{\left|a_{n+1}\right|}{\left|a_n\right|}=\frac{\frac{1}{(n+1)^2}}{\frac{1}{n^2}}=\frac{n^2}{(n+1)^2}\longrightarrow 1$.
2) As $n\to\infty$,  $\displaystyle\frac{\left|a_{n+1}\right|}{\left|a_n\right|}=\frac{\frac{1}{n+1}}{\frac{1}{n}}=\frac{n}{n+1}\longrightarrow 1$.
3) As $n\to\infty$,  $\displaystyle\frac{\left|a_{n+1}\right|}{\left|a_n\right|}=\frac{\frac{1}{n+1}}{\frac{1}{n}}=\frac{n}{n+1}\longrightarrow 1$.
We used other tests to determine the convergence/divergence of these series - the Ratio Test fails to help us with these series.

DO:  The only way a series could be conditionally convergent is if the Ratio Test fails for that series.  Why?

Review of simiplification

As you work through this module, you must be able to work with ratios of factorials as well as ratio of powers.  Recall that $n!=1\cdot 2\cdot 3\cdots(n-1)\cdot n$.  DO:  Simplify $\frac{(n+1)!}{n!}$.

$\frac{(n+1)!}{n!}=\frac{1\cdot 2\cdot 3\cdots(n-1)\cdot n\cdot (n+1)}{1\cdot 2\cdot 3\cdots(n-1)\cdot n}=n+1$ after all the cancellation.

DO:  Simplify $\displaystyle\frac{\frac{50^{n+1}}{(n+1)!}}{\frac{50^n}{n!}}$


A couple of worked out examples of the Ratio Test are contained in the video, as well as the ideas of why the Ratio Test works.