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SEQUENCES

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SEQUENCES

DEF:

A sequence is a list of numbers in a given order:

Example

first term

second term

n-th term

index

Example

Example

SEQUENCES

DEF:

A sequence is a list of numbers in a given order:

Example

Example

SEQUENCES

Example

Find a formula for the general term of the sequence

the digit in the th decimal place of the number pi

Recursive Definitions

Example

Find a formula for the general term of the sequence

This sequence arose when the 13th-century Italian mathematician known as Fibonacci

SEQUENCES

Representing Sequences

Example

LIMIT OF THE SEQUENCE

as

If converges to L, we write

We say the sequence

convg

Remark:

or simply

and call L the limit of the sequence

If there exist no L then we say the sequence is divergent.

Remark:

SEQUENCES

Convergence or Divergence

How to find a limit of a sequence

Example

(IF you can)

use Math-101 to find the limit.

Use other prop.

To find the limit

abs,r^n,bdd+montone

1

Example:

2

1)Sandwich Thm:

2)Cont. Func. Thm:

3

3)L’Hôpital’s Rule:

SEQUENCES

SEQUENCES

Example

Note:

SEQUENCES

Factorial;

NOTE

Example

SEQUENCES

Example

Find

where

Sol:

by sandw. limit is 0

SEQUENCES

Example

For what values of r is the sequence convergent?

SEQUENCES

SEQUENCES

DEFINITION

DEFINITION

bounded from above

bounded from below

Upper bound

Lower bound

If m is a lower bound but no number greater than m is a lower bound then m is the greatest lower bound

If M is an upper bound but no number less than M is an upper bound then M is the least upper bound.

Example

Is bounded below

Example

greatest upper bound = ??

Is bounded above by any number greater than one

If is bounded

from above and below,

If is not bounded

we say that

unbounded

bounded

Least upper bound

SEQUENCES

If is bounded

from above and below,

If is not bounded

we say that

unbounded

bounded

Example:

unbounded

bounded

SEQUENCES

DEFINITION

DEFINITION

non-decreasing

non-increasing

Example

Sol_2

Sol_1

Is the sequence inc or dec

SEQUENCES

DEFINITION

DEFINITION

non-decreasing

non-increasing

Example

Is the sequence inc or dec

SEQUENCES

DEFINITION

non-decreasing

DEFINITION

non-increasing

DEFINITION

monotonic

if it is either nonincreasing or nondecreasing.

SEQUENCES

1) bounded

THM6

convg

2) monotonic

THM_part1

THM_part2

non-decreasing

non-increasing

convg

convg

bounded by above

bounded by below

SEQUENCES

1) bounded

THM6

convg

2) monotonic

Example

Is the sequence inc or dec

SEQUENCES

How to find a limit of a sequence (convg or divg)

(IF you can)

use Math-101 to find the limit.

Use other prop.

To find the limit

abs,r^n,bdd+montone

Example:

Example:

1)Sandwich Thm:

1)Absolute value:

2)Cont. Func. Thm:

2)Power of r:

3)L’Hôpital’s Rule:

3)bdd+montone:

Bdd + monton convg

SEQUENCES

SEQUENCES

SEQUENCES

SEQUENCES

SEQUENCES

SEQUENCES

SEQUENCES

SEQUENCES

TERM-082

SEQUENCES

TERM-082

SEQUENCES

TERM-092

SEQUENCES

TERM-092

SEQUENCES

If is bounded

from above and below,

If is not bounded

we say that

unbounded

bounded

Example:

unbounded

bounded