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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 Root Test

The Root Test involves looking at $\displaystyle\lim_{n\to\infty}\sqrt[n]{\left|a_n\right|}$, hence the name. 

Notice:  $\displaystyle\sqrt[n]{\left|a_n\right|}=\left|a_n\right|^{1/n}$, and you will see both notations.

The Root Test, like the Ratio Test, is a test to determine absolute convergence (or not).  While the Ratio Test is good to use with factorials, since there is that lovely cancellation of terms of factorials when you look at ratios, the Root Test is best used when there are terms to the $n^{th}$ power with no factorials.

The Root Test:  Suppose that $\displaystyle\lim_{n \to \infty}\sqrt[n]{\left|a_n\right|}=L$.

  • 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 $L$ does not exist, then the test is inconclusive and we say the Ratio Test fails.  We must do more work to determine convergence, 

Example:  Test the absolute/conditional convergence of the series $\displaystyle\sum_{n=1}^\infty(-1)^n\left(\frac{2n-7}{5n+2}\right)^n$.

Solution:  We run through our tests:  Is this a geometric series?  No, we have a function raised to the $n^{th}$ power, not a number (but we get a glimmer of something here, right?).  Our terms are alternating, but the AST will not tell us whether or not we have absolute convergence.  We don't want to think about integrating this expression.  We could use the Ratio Test, but since our terms are raised to the $n^{th}$ power, we decide to try the Root Test.  You will see how nicely it works with powers:
As $n\to\infty$, $\displaystyle\sqrt[n]{\left|a_n\right|}=\sqrt[n]{\left(\frac{2n-7}{5n+2}\right)^n}=\frac{2n-7}{5n+2}=\frac{2-\frac7n}{5+\frac2n}\longrightarrow\frac25$.  Since $\frac25<1$, our series converges absolutely.

It is important to consider our litany of convergence/divergence tests before doing work.  It can save valuable time, as well as helping you begin to recognize which test to do when.  On an exam, you will not know which module the series came from!

An important limit you may need in order to use the Root Test

You may have to compute the limit of sequences like $\sqrt[n]n$ or $\lim\sqrt[n]{n^2}$.  Let's compute this now, so we can use it later.  Let $a$ be a positive real number.   First notice that $\displaystyle\lim_{n\to\infty}\sqrt[n]{n^a}=\lim_{n\to\infty}\left(n^a\right)^{1/n}=\lim_{n\to\infty}n^{a/n}$ looks like the indeterminate form $\infty^0$, so it can best be computed by computing the limit of the logarithm of this expression.  We will use the function with continuous variable $x$ so that we can take derivatives using l'Hospital's Rule.

$\displaystyle\lim_{x\to\infty}\ln\left(x^{a/x}\right)=\lim_{x\to\infty}\frac ax\cdot\ln x \underset{\,\\ 0\cdot\infty}{=}\lim_{x\to\infty}\frac{\ln x}{\frac xa}\underset{ \,\\ \frac{\infty}{\infty},\text{ l'H}}{=}\lim_{x\to\infty}\frac{\frac 1x}{\frac 1a}=\lim_{x\to\infty}\frac ax=0$,
so $\displaystyle\lim_{x\to\infty}x^{a/x}=e^0=1$.  We have shown:

A Handy Limit

For any positive real number $a$, $$\lim_{n\to\infty}\sqrt[n]{n^a}=\lim_{n\to\infty}n^{a/n}=1.$$

Justification of the Root Test

Notice that we can think of the relationship of $\sqrt[n]{\left|a_n\right|}$ to the terms of a geometric series, since $\sqrt[n]{r^n} = r$.  The video gives a justification, using this idea, of why the Root Test works.