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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
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TERM-082
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TERM-082
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TERM-092
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TERM-092
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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
