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Sec 11.1: SEQUENCES
DEF:
A sequence is a list of numbers written in a definite 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
n
n
1
n 1
n
n 1
3 n 2
1
n
Sec 11.1: SEQUENCES
DEF:
A sequence is a list of numbers written in a definite 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
Sec 11.1: 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
Sec 11.1: SEQUENCES
PLOT THE SEQUENCES
Example
1 2 3 4
, , , ,
2 3 4 5
n
n
1
n 1
Sec 11.1: 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
Sec 11.1: SEQUENCES
Sec 11.1: SEQUENCES
Sec 11.1: SEQUENCES
Example
n 1
lim (1)
n
n
Note:
n
n
lim (1)
n
n 1
Sec 11.1: SEQUENCES
Example Determine whether the sequence is convergent or divergent.
lim (1)
n
n
Sec 11.1: SEQUENCES
THEOREM;
f (x ) continuous
an convergent
an L
f (an ) convergent
f (an ) f ( L)
Example
Find
lim sin( n )
n
Sec 11.1: SEQUENCES
THEOREM;
(SQUEEZE THEOREM FOR SEQUENCES)
an bn cn for n n0
an L
cn L
3! 3x2x1 6
5! 5x4x3x2x1 120
Example
Find
where
n!
lim n
n n
n! 1 2 3 (n 1) n
bn L
Sec 11.1: 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
Sec 11.1: SEQUENCES
n
The sequence { r } is
conv
div
1 r 1
other valu es
Sec 11.1: SEQUENCES
DEFINITION
{ an }
increasing
an an1 for all n 1
a1 a2 a3 a4
DEFINITION
{ an }
decreasing
an an1 for all n 1
a1 a2 a3 a4
Sec 11.1: SEQUENCES
DEFINITION
{ an }
increasing
an an1 for all n 1
a1 a2 a3 a4
DEFINITION
{ an }
decreasing
an an1 for all n 1
a1 a2 a3 a4
DEFINITION
{ an }
monotonic
if it is either increasing or decreasing.
Sec 11.1: SEQUENCES
DEFINITION
{ an }
increasing
an an1 for all n 1
DEFINITION
{ an } 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
SYLLABUS
Sequences (up to page 682 only, End of example 12)
Sec 11.1: 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
Sec 11.1: SEQUENCES
How to find a limit of a sequence
(IF you can)
Convert into
function and use
Math-101 to find
the limit.
Example:
n
lim
n n 1
Use other prop. To find the
limit
bn an cn
conv
{ r n } is
div
squeeze
1 r 1
other valu es
an 0 an 0
Example:
n!
lim n
n n
Sec 11.1: SEQUENCES
Sec 11.1: SEQUENCES
Sec 11.1: SEQUENCES
Sec 11.1: SEQUENCES
Sec 11.1: SEQUENCES
Sec 11.1: SEQUENCES
Sec 11.1: SEQUENCES
Sec 11.1: SEQUENCES
DEFINITION
{ an } bounded above
if there is a number M such that
an M for all n 1
Example
Is the sequence bounded above
or below
n
n 1
DEFINITION
{ an } bounded below
if there is a number m such that
m an for all n 1
Sec 11.1: SEQUENCES
DEFINITION
DEFINITION
DEFINITION
{ an } bounded above
an M for all n 1
{ an } bounded below m an for all n 1
{ an }
bounded
If it is bounded above and below
Example
Is the sequence bounded
n
n 1
Sec 11.1: SEQUENCES
THEOREM;
an
an
increasing
bounded above
an
convergent
Sec 11.1: SEQUENCES
THEOREM;
an
an
increasing
bounded above
THEOREM;
an
an
decreasing
bounded below
an
convergent
an
convergent
THEOREM;
Every bounded, monotonic sequence is convergent
NOTE: The theorem doesn’t tell us what the value of the limit is.
Sec 11.1: SEQUENCES
Sec 11.1: SEQUENCES
TERM-082
Sec 11.1: SEQUENCES
TERM-082
Sec 11.1: SEQUENCES
TERM-092
Sec 11.1: SEQUENCES
TERM-092
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