theorem 1-a let be a sequence: l l a) converges to a real number l iff every subsequence of converge...
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Theorem 1-a
12
1)
1)
nii
ni
,.....1000000
1,.....,3
1,2
1,1
Let <S<Snn>> be a sequence:
a) <S<Snn>> converges to a real number LL iff every subsequence of <Sn><Sn> converge to LL
Illustrations
,......1000001
1,.....,9
1,7
1,5
1,3
1,1
1
00
00
Theorem 1-b
If L is a limit of <Sn><Sn>, then LL is the only limit of <Sn>.<Sn>.
IllustrationsIllustrations
mitliothernohasSnn
n
n
nSn
n1
1lim
1
Example 1
8lim
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:
lim,
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8
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Solution
ititsfindconvergessthatngAssumi
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Example 2
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ititsfindconvergessthatngAssumi
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Example 3
divergess
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ns
nn
n
nn
ns
nn
n
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Solution
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Theorem 1-c
0lim),,0[
nn
Sthen
3
5
nn
c) If <Sn><Sn> is eventually in
053
5lim
n
nn
Illustration
d) If <Sn><Sn> is eventually in 0lim,0
n
n
SthenR
2
5
n
n
052
5lim
n
nn
Illustration
Theorem 1-d
Theorem 2
1
5
nn
Let <S<Sn>> be a sequence:
a) If <S<Snn>> is bounded from above and increasing then it converge to the supremum of the range of <S<Snn>> .
Illustration
55
It is bounded from above & increasingIt converges to the sup of its range, which is 5
Question
convergesdoeswhyExplain
ss
sLet
n
nn ;
2
8
5
1
1
Question
convergesdoeswhyExplain
ss
sLet
n
nn ;12
12
1
1
Theorem 2
b) If <S<Snn>> is bounded from below and decreasing then it converges to the infimum of the range of <S<Snn>> .
n
10
Illustration
00
It is bounded from below & decreasingIt converges to the inf of its range which is 0
Theorem 3
ennn
)1(lim 1
Nnsn ;32
A convergent sequence is bounded n
n
11
Operations On Convergent Sequences
21 limlim ltlSlet nn
nn
Nntandll
l
t
Se
Nntandllt
d
lltSc
lltSb
RlSa
then
nn
n
n
nn
n
nnn
nnn
nn
;00;lim)
;00;11
lim)
lim)
)(lim)
;)(lim)
,
22
1
22
21
21
1
Illustrations
252
14limlim3
1
3limlim
52
14
1
3
2
2
2
2
n
nk
n
ns
Kn
nS
n
n
let
nnn
nnn
nn
2
3
lim
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lim
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K
s
K
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KKd
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Find
Examples
1.1
19
1.3
2
53.2
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1(1.1
2
3
3
n
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e
ee
n
n
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sequencetheofmitlitheFind
Solutions
101)5
1(lim1lim))
5
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)5
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nnn
n
n
n
Solutionss
7
5
20
50
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lim
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lim
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lim2
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2
53.2
2
3
32
3
3
3
3
3
nn
nn
nn
n
n
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nnn
n
nn
n
Solutionss
4
1
016
01
3lim16lim
1lim1lim
316
1lim
316
1lim
316
1.3
n
n
n
n
n
n
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nn
nn
nn
Solutionss
01
0
01
00
1lim1lim
1lim
1lim
11
11
lim1
lim
1.4
2
3
2
3
2
2
nnn
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Question
1
1
:
1011
aifsequencegalternatinisa
aiftodivergesa
isa
thatCheck
n
n
toConvergent
otherwiseDinvergentn
aifaif