Numerical Sequences

Why Sequences?

• There are six animations about limits to show the sequence in the domain and range. Problems displaying the data. It has to be fixed

• Intuition about Infinitely close and direction

Why do we need to approach to the limit point systematically?

The behavior of the function g(x)=1/x was previously discussed for values of x "close to zero", for values of x "very large but positive", and for values of x

"very large but negative".

Support from TablesVerify these tables with the calculator

Challenge

The graph of a function f(x) is given but you don’t know anything else about the function other than its graph and values of the function at points in the domain which are "close" to zero.

Use the graph of the function y=f(x) to make conjectures about the following values

Are these conjectures supported by the tables?

The Graph of the same function in two Different Windows

Discuss the limits below on both windows. Are they the same? Should they be the same?

Definition of Numerical Sequences

Listing– Finite – Infinite

Formulas (graph as listing and as functions)• a. • b. • c. • d.

Basic Functions vs.. Basic Sequences

• Show graphing sequences• Indicate how to start always at zero• Sequence as a listing vs. sequence as a function

Key types of sequences

• Diverging to infinity

• Converging to zero

Sequences Diverging to InfinityNumber Positive Infinitely Large “∞”

SliderNormally a LARGE

NUMBER

SliderNormally a

LARGE NUMBER

EXAMPLE

A=800

When N=29 First term

passing slider

What is N when A=1028?

Show in calculator

ExerciseFor the sequence below find the number of terms less than or equal to each of the given values of A. Represent the results geometrically in one and two dimensions. For each A determine the value N satisfying the condition

Any number within ε from 0 approximates 0 with accuracy less than ε

Any number within ε from 0 approximates 0 with accuracy less than ε

Positive Sequences Converging to ZeroNumbers Positive Infinitely Small “0+”

SLIDER ε (small) distance

from zero

SLIDERε (small) distance

from zero

Any term of the sequence here represents 0 accurate to three decimal places

EXAMPLE

When N=1001 first term passing SLIDER. The term is

a1001=1/1001 a1001=0.000999

When N=1001 first term passing SLIDER. The term is

a1001 =1/1001 a1001 =0.000999

Show in calculator

What is N when ?

EXERCISE

For the sequence find the number of terms less than or equal to the given values of .Start by construction the table of values of the sequence, and represent the results on the number line/graph. Finally support your conclusions algebraically.

Relating Negative Infinitely Large and Negative Infinitely Small Numbers

Positively Large Numbers

Positively Small Numbers

Yield

• Negative Infinitely large numbers are the opposite of Positive Large Numbers

• Negative Infinitely Small Numbers are the opposite of Positive Infinitely Small Numbers

Relating Positive Infinitely Large and Positive Infinitely Small Numbers

Negative Large Numbers

Negative Small Numbers

Yield

Converging to Any Number Other Than Zero

Generate three examples that represent each of the following expressions “7+”, “7-”, “7”

Key Observation

• “0+” are always positive• “0-” are always negative• “0” could be either “0+” or “0-”

• “7+”, “7- ”, “7” represent always positive numbers (infinitely close to 7)

“a” means numbers infinitely close to a, a any real number

Verification Time

• Provide four examples that represent each of the following expressions – “∞”– “0”– “0.2+”

– “_ 0.002-”

Operations with “ “Operation between sequences are point-wise or term by term

Notation

c ×”∞"=“∞", when c is positive

c ×”∞"=“-∞", when c is negative

Represents any sequence converging to zero from the right added to a sequence diverging to infinity

Addition

Product

Quotient

Trouble-MakersEach situation has to be analyzed. No general rules as in

the other cases

Dominance of Sequences

The sequence an DOMINATES the sequence bn as n goes to infinity means that

Exercise

Show that the sequence on the left column dominates the sequence on the second column

Exercise 4 Workbook 4.3

Two exercises assigned to each group.

Transformation of sequences under Functions

indicates the behavior of the heights of the function f(x) for values of x in the

domain infinitely close to a from the right.

The values of x could be "any sequence” converging to a from the right.

Consider the function

Exercise 2