Unit: Sequences and Series
Module: The Ratio and Root Tests
The Ratio Test
To apply the ratio test to a series
n =1
an , let
= lim
n
an +1 . an
If < 1, then the series converges absolutely. If > 1, then the series diverges. If = 1, then the test is inco

Unit: Sequences and Series
Module: Power Series Representations of Functions
Finding Power Series Representations by Differentiation
You can do calculus on a power series inside its interval of convergence. The derivative of the power series
n =0
an (

Unit: Sequences and Series
Module: Power Series Representations of Functions
Differentiation and Integration of Power Series
You can do calculus on a power series inside its interval of convergence. The derivative of the power series
n =0
an ( x c )n

Unit: Sequences and Series
Module: Power Series
Interval and Radius of Convergence
The interval of convergence of a power series is the collection of points for which the series converges. The radius of convergence of a power series is the distance betw

Unit: Sequences and Series
Module: Power Series
Definition of Power Series
The general form of a power series centered at x = c is a sequence of numbers that only depend on n.
n =0
an ( x c )n , where an is
All power series converge for x = c. If a pow

Unit: Sequences and Series
Module: Taylor and Maclaurin Series
Taylor Series
The Taylor series expansion of f ( x ) centered at x = c is
f ( n ) (c ) ( x c )n n! n =0
assuming that f ( x ) is differentiable an infinite number of times. For a given x, if

Unit: Sequences and Series
Module: Taylor and Maclaurin Polynomials
Maclaurin Polynomials
The Maclaurin polynomial of f (x ) is the Taylor polynomial of f (x ) centered at k f n0 n f (2) (0) 2 f ( k ) (0) k x = 0: x = f (0) (0) + f (1) (0)x + x +K+ x. 2 k

Unit: Sequences and Series
Module: Taylor and Maclaurin Polynomials
Taylor Polynomials
Higher-degree polynomial approximations result in more accurate representations. To find a Taylor polynomial, find all the necessary derivatives and evaluate them at

Unit: Sequences and Series
Module: Polynomial Approximations of Elementary Functions
Polynomial Approximation of Elementary Functions
Polynomials can approximate complicated functions. The tangent line approximation of f (x ) at x = c is y = f (c ) + f

The Root Test
To apply the nth root test to a series
n =1
an , let
= lim
n
n
an .
If < 1, then the series converges absolutely. If > 1, then the series diverges. If = 1, then the test is inconclusive. The nth root test is another way to see if the terms

Unit: Sequences and Series
Module: Power Series Representations of Functions
Finding Power Series Representations by Integration
You can do calculus on a power series inside its interval of convergence. The integral of the power series
n =0
an ( x c )n

Unit: Sequences and Series
Module: Infinite Series
[page 1 of 2]
Introduction to Infinite Series
Binary operations combine two values to yield a single result. You can add as many numbers as you want as long as you only have finitely many of them. Since

[page 1 of 2]
Monotonic and Bounded Sequences
A sequence is increasing if each term is larger than the previous one. A sequence is decreasing if each term is smaller than the previous one. A sequence is monotonic if the terms are nonincreasing (a1 a2 an

Unit: Sequences and Series
Module: Sequences
The Limit of a Sequence
A sequence is a pattern of numbers. If a sequence approaches a particular number as the index n gets larger, then the sequence is said to converge to that number. If not, the sequence

Unit: Applications of Integration
Module: Work
Introduction to Work
Work is the energy used when applying a force over a distance. For a constant force F, work is the product of the force and the change in distance. For a changing or variable force F(x)

Unit: Applications of Integral Calculus
Module: Arc Length
Introduction to Arc Length
Arc length is the length of the curve. The arc length of a smooth curve given by the function f (x) between a and b:
b a
1 + [f ( x )]2 dx .
W hen measuring how long a

Unit: Applications of Integration
Module: Disks and Washers
Solids of Revolution
Revolving a plane region about a line forms a solid of revolution. The volume of a solid of revolution using the disk method where R (x) is the radius of the solid of revol

Unit: Applications of Integration
Module: Finding Volumes Using Cross-Sections
Finding Volumes Using Cross-Sectional Slices
The volume of a solid with vertical cross-sections of area A (x) is V where:
V=
b
A( x ) dx .
a
The volume of a solid with hori

Unit: Improper Integrals
Module: Convergence and Divergence of Improper Integrals
The Second Type of Improper Integral
If a function is not continuous on the integration interval, then the standard procedure will not work. Use the discontinuity as an endp

Unit: Improper Integrals
Module: Convergence and Divergence of Improper Integrals
The First Type of Improper Integral
An improper integral is a definite integral with one of the following properties: the integration takes place over an infinite interval

Unit: Techniques of Integration
Module: Trigonometric Substitution Strategy
An Overview of Trig Sub Strategy
Use trigonometric substitution to evaluate integrals involving the square root of the sum or difference of two squares. 1. Match the square root e

Unit: Techniques of Integration
Module: Introduction to Trigonometric Substitution
Trigonometric Substitutions on Rational Powers
Use trigonometric substitution to evaluate integrals involving the square root of the sum or difference of two squares. 1. Ma

Unit: Techniques of Integration
Module: Introduction to Trigonometric Substitution
Using Trigonometric Substitution to Integrate Radicals
Use trigonometric substitution to evaluate integrals involving the square root of the sum or difference of two square

Unit: Elementary Functions and Their Inverses
Module: Calculus of Inverse Trig Functions
Derivatives of Inverse Trigonometric Functions
To find the derivative of an inverse trig function, rewrite the expression in terms of standard trig functions, differe

Unit: Elementary Functions and Their Inverses
Module: Inverse Trigonometric Functions
The Inverse Sine, Cosine, and Tangent Functions
The standard trigonometric functions do not have inverses. Only by restricting the domain can you make them one-to-one

Elementar y Functions and Their Inverses
Derivatives of Inverse Functions
You can calculate the derivative of an inverse function at a point without determining the actual inverse function. The inverse of a function retains many of the properties of the

Unit: Elementary Functions and Their Inverses
Module: Inverse Functions
The Basics of Inverse Functions
Inverse functions undo each other. In inverse functions, the dependent variable and independent variable switch roles. The graph of an inverse functi

Unit: LHpitals Rule
Module: Other Indeterminate Forms
Another Example of One to the Infinite Power
Some indeterminate forms have to be transformed before you can apply LHpitals rule. In order to apply LHpitals rule to a limit of the form 1 use the prope

Unit: LHpitals Rule
Module: Indeterminate Quotients
[page 1 of 1]
An Intro to LHpitals Rule
A limit of a function is called an indeterminate form when it produces a mathematically meaningless expression. Indeterminate forms are also called indeterminant

Unit: Sequences and Series
Module: The Limit Comparison Test
[page 1 of 2]
Introduction to the Limit Comparison Test
The limit comparison test: Consider two series bn > 0, and lim diverge.
n =1
a n and b n , with an > 0,
n =1
an
n
n b
= L . If 0 < L < ,