lecture 7: taylor series validity, power series, vectors, dot products, and cross product

Lecture 7: Taylor Series Validity, Power Series, Vectors, Dot

Products, and Cross Product

Part I: Validity of Taylor Series

Taylor Series Failure

• Consider the function if , if .• for all n, so the Taylor series is just 0

Taylor series remainder term• Recall that • When the equation fails, it is because of a

“remainder term” at infinity• Similarly, • so

• This process can be continued.

Part II: Power Series

Objectives

• Know what power series are and how to determine the radius of convergence of power series

• Know how power series can be manipulated within their radius of convergence

Corresponding sections of Simmons: 14.2,14.3

Power Series

• Def: A power series is a series of the form • Examples:

• Taylor Series around

Radius of Convergence• The radius of convergence of a power series is

the number such that converges if and diverges if (the behavior at is undertermined)

• Example: For , • Some power series, such as , converge for all .

In this case, .• Some power series, such as , diverge for all . In

this case, .

Manipulating Power Series• As long as we are within the radius of convergence

of a power series, we can differentiate, integrate, and make substitutions in power series.

• Cool example:• for .• Plugging in for , for • Integrating gives for

Uniqueness of Power/Taylor Series• Fact: The Power/Taylor series for a given

function is unique.• Reason: Completely determined by the fact

that it has to match the nth derivative of the function

• Corollary: If we find a power series for a function by any means, it must be valid!

• Plugging in ,

Part III: Vectors

Objectives

• Know what vectors are• Know how find the magnitude, direction, dot

product, and cross product of vectors.Corresponding sections in Simmons: 18.1, 18.2, 18.3

Vectors

• A vector is described by n coordinates .• We write• The magnitude of , written as , is

Picture for vectors

• A vector can be thought of as an arrow.

• is the length of the arrow

0 1 2 3 4-1-2-3-4

01234

-1-2-3-4

5 6-5-6

-5-6

56

𝑣=¿3,5>¿

Scalar Multiplication of Vectors and Vector Direction

• If then • The direction of a vector , denoted as , is • is the vector with length 1 which points in the

same direction as .

Vector Addition and Subtraction

• If and then • If and then

Displacement and Position Vectors

• Given points and , • The position vector is the displacement vector

from the origin.• If is the origin and then • Proposition: Given points P,Q,R,

Dot Product

• If and then • Example:

Properties of the dot product

• Linearity in and :• Commutativity:

Geometric Picture of the dot product

• , where is the angle between and .• Note that is the length of the projection of

onto

𝜃𝑣

�⃗�

projection of onto

Connection between algebraic and geometric pictures

• Law of cosines:

a

bc A

B C

• Proof: and algebra gives the result• Algebraically,

|�⃗�+𝑤|2=(�⃗�+�⃗� ) ∙ (�⃗�+�⃗� )=|⃗𝑣|2+|�⃗�|2+2 �⃗� ∙�⃗�• Take and

P Q

R

Condition for Perpendicular Vectors

• Corollary: if and only if

Cross Product• The cross product is defined for 3 dimensions.• Definition: If , , and then

• Example:

Properties of the cross product

• Linearity in and :• Anti-Commutativity:• Warning:

Geometric Picture of the Cross Product

• has magnitude equal to the area of the parallelogram with sides and

• is perpendicular to both and • Right-hand rule: To find the direction of , point

your right thumb in the direction of . Then point your other fingers in the direction of . will point in the direction which goes outward from your palm.