SEQUENCES A sequence is a list of numbers in a given order: DEF: Example first termsecond term n-th...

SEQUENCES

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

, , , , 21 naaa

DEF:

,5

1 ,

4

1 ,

3

1 ,

2

1 ,1

Example

,5

4 ,

4

3 ,

3

2 ,

2

1

,81

5 ,

27

4 ,

9

3 ,

3

2

Example

Example

first term second term n-th term

1

1

nn nna 1

11 nn

n

213 n

n

n

index

SEQUENCES

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

, , , , 21 naaa

DEF:

,, , , ,

6

5

5

4

4

3

3

2

2

1

,

81

5 ,

27

4 ,

9

3 ,

3

2

Example

Example

11

nn

n

21

1

3)1(

nn

n n

n)1(

SEQUENCES

Find a formula for the general term of the sequenceExample

,6 ,2 ,9 ,5 ,1 ,4 ,1

Find a formula for the general term of the sequenceExample

21 ,31 ,8 ,5 ,3 ,2 ,1 ,1

3589793.14159265

the digit in the th decimal place of the number pi

21

2

1 ,1 ,1

nnn fffff

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

Recursive Definitions

SEQUENCES

,5

4 ,

4

3 ,

3

2 ,

2

1

Example

11 nn

n

Representing Sequences

LIMIT OF THE SEQUENCE

11n

nnas

11

lim n

nn

We say the sequence 1n

nna convg

Rem

ark:

If converges to L, we write 1nna

Lann

lim

or simply

and call L the limit of the sequence

Lan Remark: If there exist no L then we say the

sequence is divergent.

SEQUENCES

Example

11 nn

n

Convergence or Divergence

12 nn

,1,1,1,1

1

2

3

How to find a limit of a sequence

Example:

1lim

n

nn 1

lim x

xx

(IF you can)

use Math-101 to find the limit.

Use other prop.

To find the limit

abs,r^n,bdd+montone

1)Sandwich Thm:

n

ncos n

n 1)1(

2)Cont. Func. Thm:

n

n 1

n

1

2

)()( LfafLa nn

3)L’Hôpital’s Rule:

n

nln

n

n

n

1

1

SEQUENCES

SEQUENCES

Example

nn

n

1)1(lim

Note:

1)1(lim

n

nn

n

SEQUENCES

Factorial;

!3!5

nnn )1(321!

Example

)!9(10!10 NOTE

)!1(! nnn

6123 12012345

SEQUENCES

Example

nn n

n!lim

Find

nnn )1(321!where

nnnn

n

n

nn

321!

0

n

n

nnnn

nn

321!

0

n

n

nnnn

nn

1321!0

one than less

nn

nn

1!0

Sol:

by sandw. limit is 0

SEQUENCES

Example

} { nr

For what values of r is the sequence convergent?

n

nr

lim

esother valudiv

11conv is } { sequence The

rr n

SEQUENCES

esother valudiv

11conv is } { sequence The

rr n

SEQUENCES

DEFINITION

} { na bounded from above

nMan allfor

Example

1n

n Is bounded above by any number greater than one

1.1na 001.1na

Upper boundM

Least upper bound1M

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

DEFINITION

} { na bounded from below

nMan allfor

Example

n

13 Is bounded below

3na

Lower boundM

greatest upper 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 is bounded from above and below,

na If is not bounded

bounded na

na

we say that

unbounded na

SEQUENCES

If is bounded from above and below,

na If is not bounded

bounded na

na

we say that

unbounded na

1n

n

n

13 2n

Example:

bounded unbounded

SEQUENCES

DEFINITION

} { na non-decreasing

1 allfor 1 naa nn

4321 aaaa

DEFINITION

} { na non-increasing

1 allfor 1 naa nn

4321 aaaa

Example

13

n

Is the sequence inc or dec

Sol_1

nn aann

nn

nn

1

13

1

13

1

1

1

1 Sol_2

)1( 01

1

2)('

3)(

xx

x

xf

xf

SEQUENCES

DEFINITION

} { na non-decreasing

1 allfor 1 naa nn

4321 aaaa

DEFINITION

} { na non-increasing

1 allfor 1 naa nn

4321 aaaa

Example

Is the sequence inc or dec

1

2n

n

SEQUENCES

if it is either nonincreasing or nondecreasing.

DEFINITION } { na monotonic

DEFINITION

} { na non-decreasing 1 allfor 1 naa nn

4321 aaaa

DEFINITION

} { na non-increasing 1 allfor 1 naa nn

4321 aaaa

SEQUENCES

THM_part1

} { na non-decreasing

bounded by aboveconvg

THM6 } { na 1) bounded

2) monotonicconvg

THM_part2

} { na non-increasing

bounded by belowconvg

SEQUENCES

THM6 } { na 1) bounded

2) monotonicconvg

Example

13

n

Is the sequence inc or dec

SEQUENCES

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

Example:

1lim

n

nn 1

lim x

xx

(IF you can)

use Math-101 to find the limit.

Use other prop.

To find the limit

abs,r^n,bdd+montone

1)Sandwich Thm:

n

ncos n

n 1)1(

2)Cont. Func. Thm:

n

n 1

n

1

2

)()( LfafLa nn

3)L’Hôpital’s Rule:

n

nln

n

n

n

1

1

1)Absolute value:

0 then 0 nn aa

2)Power of r:

3)bdd+montone:

Bdd + monton convg

Example:n)1( !n

SEQUENCES

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SEQUENCES

TERM-082

SEQUENCES

TERM-082

SEQUENCES

TERM-092

SEQUENCES

TERM-092

SEQUENCES

If is bounded from above and below,

na If is not bounded

bounded na

na

we say that

unbounded na

1n

n

n

13 2n

Example:

bounded unbounded