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SEQUENCES
DEF:
A sequence is a list of numbers in a given order:
a1 ,
Example
first term
Example
Example
a2 , , an , 
1 1 1 1
1, , , , , 
2 3 4 5
second term
1 2 3 4
, , , , 
2 3 4 5

1 
 
 n  n 1
an 
n-th term
1
n
index

 n 


n

1

n 1

2 3 4
5
n

1


, ,
,
,   n 
3 9 27 81
 3 n 1

OR
 n 
 n 1 
 3 n  2
SEQUENCES
DEF:
A sequence is a list of numbers in a given order:
a1 ,
a2 , , an , 
Example 
1 2
3 4
5


,
,

,
,

,





4 5
6
 2 3

Example 
2 3
4
5


,
,

,
,





27 81
 3 9


(1) n n 


n

1

n 1


n 1 n 
(

1
)

n 1 
3

n  2
SEQUENCES
Find the n-term
TERM-092
1 
 2n
an  


 n  3 2n  1 
n  1,2,3, 
TERM-131
an 
(n)( n  2)
(n  1)
n  1,2,3, 
SEQUENCES
Example
Find a formula for the general term of the sequence
  3.14159265358979
1, 4, 1, 5, 9, 2, 6,
the digit in the th decimal place of
the number pi
Recursive Definitions
Example
1, 1,
Find a formula for the general term of the sequence
2, 3, 5, 8, 13, 21
a1  1,
a2  1,
an  an 1  an  2
This sequence arose when the 13th-century Italian mathematician known as
Fibonacci
SEQUENCES
Representing Sequences
Example

1 2 3 4
, , , , 
2 3 4 5
 n 


n

1

n 1
LIMIT OF THE SEQUENCE
as
Remark:
If
an n1
n 
We say the sequence
n 
or simply
an  L
and call L the limit of the sequence
1
n
lim
1
n  n  1
converges to L, we write
lim an  L
n
n 1
Remark:
an 
n
n 1
convg
If there exist no L then we say the
sequence is divergent.
SEQUENCES
Convergence or Divergence
Example

1
2
 n 


 n  1n 1
2 
n 
n 1
How to find a limit of a sequence
(IF you can)
Use other prop.
use Math-101
to find the limit.
To find the limit
abs,r^n,bdd+montone
Example:
n
n n  1
lim
lim
x
x
x 1
1)Sandwich Thm:
  
cos n
n
(1)
n 1
n

2)Cont. Func. Thm:
3 1,1,1,1,
an  L  f (an )  f ( L)
 
n 1
n
 1n 
2 
 
3)L’Hôpital’s Rule:
 ln n 


 n 
n

 n  1  

 


 n  1  

SEQUENCES
SEQUENCES
Example

n 1
lim  (1) 
n 
n

Note:
n 

n
lim  (1)

n 
n 1

SEQUENCES
Factorial;
n! 1 2  3    (n  1)  n
Example
3! 3 2 1  6
5! 5  4  3 2 1  120
NOTE
10! 10  (9!)
n! n  (n  1)!
SEQUENCES
Example
Find
where
Example
n! 1 2  3    (n  1)  n
Find
where
1
lim
n   n!
9n 2  5
lim
n 
n!
n! 1 2  3    (n  1)  n
SEQUENCES
Example
For what values of r is the sequence convergent?
n
{r }
n
The sequence { r } is
lim r
n
n
conv

 div
1  r  1
other valu es
SEQUENCES
1  r  1
other valu es
conv

 div
n
The sequence { r } is
Example:
9.7 
n
  1 
n
 0.99 
0.5 
n
n
 1 
n
SEQUENCES
DEFINITION
{ an }
DEFINITION
{ an }
bounded from above
an  M for all n
M
Upper bound
an  M for all n
M
If M is an upper bound but
no number less than M is
an upper bound then M is
the least 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
Lower bound
Example 3  1  Is bounded below


Example
 n 


 n  1
bounded from below

Is bounded above
by any number
greater than one
an  1.1
an  1.001
M 1
Least upper bound
n
If an  is bounded
from above and below,

an 
bounded
an  3
If
an 

greatest upper
bound = ??
is not bounded
we say that
an 
unbounded
SEQUENCES
If an  is bounded
from above and below,

an 
Example:
If
an 

is not bounded
we say that
an 
bounded
 n  3  1 

  n
 n  1
bounded
unbounded
n 
2
unbounded
SEQUENCES
DEFINITION
DEFINITION
{ an }non-decreasing
{ an }non-increasing
an  an1 for all n  1
an  an1 for all n  1
a1  a2  a3  a4  
Example
1 

3



n 

Is the sequence inc or dec
a1  a2  a3  a4  
n 1  n
1
1

n 1 n
1
1
3
 3
n 1
n
an 1  an
Sol_1
Sol_2
f ( x)  3  1x
f ' ( x)   x12
 0
( x  1)
SEQUENCES
DEFINITION
DEFINITION
{ an }non-decreasing
{ an }non-increasing
an  an1 for all n  1
a1  a2  a3  a4  
Example
 n 
 2

 n 1 
Is the sequence inc or dec
an  an1 for all n  1
a1  a2  a3  a4  
SEQUENCES
DEFINITION
{ an }
non-decreasing
an  an1 for all n  1
a1  a2  a3  a4  
DEFINITION
{ an } non-increasing an  an1 for all n  1
a1  a2  a3  a4  
DEFINITION
{ an }
monotonic
if it is either nonincreasing or nondecreasing.
SEQUENCES
DEFINITION
{ an }
non-decreasing
an  an1 for all n  1
a1  a2  a3  a4  
Is the sequence non-decreasing?
SEQUENCES
THM6 { an }
1) bounded
convg
2) monotonic
THM_part1
{ an } non-decreasing
bounded by above
THM_part2
convg
{ an } non-increasing
bounded by below
convg
SEQUENCES
THM6 { an }
1) bounded
convg
2) monotonic
Example
1 

3



n 

Is the sequence inc or dec
SEQUENCES
How to find a limit of a sequence (convg or divg)
(IF you can)
Use other prop.
use Math-101
to find the limit.
To find the limit
abs,r^n,bdd+montone
Example:
n
lim
n n  1
Example:
x
lim
x x  1
1)Sandwich Thm:
  
cos n
n
(1)
n 1
n

(1) n
n!
1)Absolute value:
an  0 then an  0
2)Cont. Func. Thm:
an  L  f (an )  f ( L)
 
n 1
n
 1n 
2 
 
3)L’Hôpital’s Rule:
 ln n 


 n 
2)Power of r:
n

 n  1  

 


 n  1  

3)bdd+montone:
Bdd + monton  convg
SEQUENCES
SEQUENCES
SEQUENCES
TERM-082
SEQUENCES
TERM-082
SEQUENCES
TERM-092
SEQUENCES
TERM-092
SEQUENCES
Example
n!
lim n
n n
Find
where
n! 1 2  3    (n  1)  n
Sol: 0  n!  1 2  3   n
n n n  n  n   n
n! 1 2 3 n
0 n 

n
nnn n
n! 1  2 3 n  1 
0 n  


n
nn n
n 


less than one
0
n! 1

n
n
n
by sandw. limit is 0
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Multiple-Choice
Problems
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If an  is bounded
from above and below,

an 
Example:
If
an 

is not bounded
we say that
an 
bounded
 n  3  1 

  n
 n  1
bounded
unbounded
n 
2
unbounded