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
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1lim
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Example 1
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Solution
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Example 2
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Example 3
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Solution
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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
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
Operations On Convergent Sequences
21 limlim ltlSlet nn
nn
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then
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Illustrations
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Find
Solutionss
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lim2
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Solutionss
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