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SEQUENCES
DEF:
A sequence is a list of numbers in a given order:
a1 ,
Example
first term
a2 , , an ,
1 1 1 1
1, , , , ,
2 3 4 5
second term
Example
1 2 3 4
, , , ,
2 3 4 5
Example
2 3 4
5
, ,
,
,
3 9 27 81
1
n n 1
an
n-th term
index
n
n
1
n 1
n
n 1
3 n 2
1
n
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
Example
(1) n n
n
1
n 1
n 1 n
(
1
)
n 1
3
n 2
Find a formula for the general term of the sequence
13 15
17
11
,
,
,
,
25 125
625
5
SEQUENCES
Example
Find a formula for the general term of the sequence
13 15
17
11
,
,
,
,
25 125
625
5
Example
Find a formula for the general term of the sequence
3.14159265358979
1, 4, 1, 5, 9, 2, 6,
Example
1, 1,
the digit in the th decimal place of
the number pi
Find a formula for the general term of the sequence
2, 3, 5, 8, 13, 21
f1 1,
f 2 1,
f n f n 1 f n 2
This sequence arose when the 13th-century Italian mathematician known as
Fibonacci
SEQUENCES
Recursive Definitions
Example
1, 1,
Find a formula for the general term of the sequence
2, 3, 5, 8, 13, 21
f1 1,
f 2 1,
f n f n 1 f n 2
This sequence arose when the 13th-century Italian mathematician known as
Fibonacci
Example
a1 1,
a1 1,
an an1 1
an n an1
SEQUENCES
PLOT THE SEQUENCES
Example
1 2 3 4
, , , ,
2 3 4 5
n
n
1
n 1
SEQUENCES
LIMIT OF THE SEQUENCES
Example
Example
1 2 3 4
, , , ,
2 3 4 5
lim an 1
n
n
n
1
n 1
n
lim
1
n n 1
SEQUENCES
Convergence or Divergence
Example
n
n 1n 1
2
n
n 1
1,1,1,1,1,1,1,
SEQUENCES
SEQUENCES
Example Determine whether the sequence is convergent or divergent.
lim (1)
n
n
SEQUENCES
Example
n 1
lim (1)
n
n
Note:
n
n
lim (1)
n
n 1
SEQUENCES
THEOREM;
f (x ) continuous
an convergent
an L
f (an ) convergent
f (an ) f ( L)
Example
Find
lim sin( n )
n
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
THEOREM;
(SQUEEZE THEOREM FOR SEQUENCES)
an bn cn for n n0
an L
cn L
Example
bn L
Find
cos n
lim
n
n
n 1
lim (1)
n
n
1
lim n
n 2
SEQUENCES
THEOREM;
(SQUEEZE THEOREM FOR SEQUENCES)
an bn cn for n n0
an L
cn L
Example
Find
where
n!
lim n
n n
n! 1 2 3 (n 1) n
bn L
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
n
The sequence { r } is
conv
div
1 r 1
other valu es
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
bounded
If
an
is not bounded
we say that
an
unbounded
SEQUENCES
DEFINITION
{ an }
non-decreasing
an an1 for all n 1
a1 a2 a3 a4
DEFINITION
{ an } non-increasing an an1 for all n 1
a1 a2 a3 a4
SEQUENCES
DEFINITION
{ an }
non-decreasing
an an1 for all n 1
a1 a2 a3 a4
DEFINITION
{ an } non-increasing an an1 for all n 1
a1 a2 a3 a4
DEFINITION
{ an }
monotonic
if it is either nonincreasing or nondecreasing.
SEQUENCES
DEFINITION
{ an }
non-increasing
an an1 for all n 1
DEFINITION
{ an }
non-decreasing
an an1 for all n 1
Example
Is the sequence increasing or decreasing
Example12 Is the sequence increasing or decreasing
2-solutions
3
n5
n
2
n 1
SEQUENCES
SEQUENCES
How to find a limit of a sequence
THEOREM;
f (x ) continuous
an convergent an L
THEOREM;
(SQUEEZE THEOREM)
an bn cn for n n0
cn L
an L
conv
The sequence { r n } is
div
1 r 1
other valu es
f (an )
convergent
f (an ) f ( L)
THEOREM;
bn L
Every bounded, monotonic
sequence is convergent
SEQUENCES
How to find a limit of a sequence
(IF you can)
use Math-101 to
find the limit.
Use other prop. To find the
limit
bn an cn
conv
{ r n } is
div
Example:
n
lim
n n 1
squeeze
1 r 1
other valu es
an 0 an 0
Example:
n!
lim n
n n
SEQUENCES
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TERM-082
SEQUENCES
TERM-082
SEQUENCES
TERM-092
SEQUENCES
TERM-092
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