SEQUENCES A sequence is a list of numbers in a given order: DEF: Example first termsecond term n-th...
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Transcript of SEQUENCES A sequence is a list of numbers in a given order: DEF: Example first termsecond term n-th...
![Page 1: SEQUENCES A sequence is a list of numbers in a given order: DEF: Example first termsecond term n-th term index.](https://reader033.fdocuments.us/reader033/viewer/2022051620/56649ee15503460f94bf2527/html5/thumbnails/1.jpg)
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
![Page 2: SEQUENCES A sequence is a list of numbers in a given order: DEF: Example first termsecond term n-th term index.](https://reader033.fdocuments.us/reader033/viewer/2022051620/56649ee15503460f94bf2527/html5/thumbnails/2.jpg)
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(
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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
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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.
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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
![Page 6: SEQUENCES A sequence is a list of numbers in a given order: DEF: Example first termsecond term n-th term index.](https://reader033.fdocuments.us/reader033/viewer/2022051620/56649ee15503460f94bf2527/html5/thumbnails/6.jpg)
SEQUENCES
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SEQUENCES
Example
nn
n
1)1(lim
Note:
1)1(lim
n
nn
n
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SEQUENCES
Factorial;
!3!5
nnn )1(321!
Example
)!9(10!10 NOTE
)!1(! nnn
6123 12012345
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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
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SEQUENCES
Example
} { nr
For what values of r is the sequence convergent?
n
nr
lim
esother valudiv
11conv is } { sequence The
rr n
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SEQUENCES
esother valudiv
11conv is } { sequence The
rr n
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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
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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
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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
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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
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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
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SEQUENCES
THM_part1
} { na non-decreasing
bounded by aboveconvg
THM6 } { na 1) bounded
2) monotonicconvg
THM_part2
} { na non-increasing
bounded by belowconvg
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SEQUENCES
THM6 } { na 1) bounded
2) monotonicconvg
Example
13
n
Is the sequence inc or dec
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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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SEQUENCES
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SEQUENCES
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SEQUENCES
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SEQUENCES
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SEQUENCES
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SEQUENCES
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SEQUENCES
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SEQUENCES
TERM-082
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SEQUENCES
TERM-082
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SEQUENCES
TERM-092
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SEQUENCES
TERM-092
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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