M38 Lec 111913
of 98
Transcript of M38 Lec 111913
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MATH 38Mathematical Analysis III
I. F. Evidente
IMSP (UPLB)
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Previously
SequencesGraph of SequencesConvergence of a SequenceTheorems for finding limits of sequences
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Outline
1 Convergence of Alternating Sequences
2 Monotone and Bounded Sequences
3 Proving the convergence of sequences by definition
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Outline
1 Convergence of Alternating Sequences
2 Monotone and Bounded Sequences
3 Proving the convergence of sequences by definition
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Alternating SequencesIf the signs of a sequence alternate from positive to negative, or vice-verse,we say that the sequence is an alternating sequence.
Example
1
{(1)n2n
}={12,1
4,1
8,1
16, ...
}2 {(1)n+1}= {1,1,1,1,1, ...}
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Alternating SequencesIf the signs of a sequence alternate from positive to negative, or vice-verse,we say that the sequence is an alternating sequence.
Example
1
{(1)n2n
}={12,1
4,1
8,1
16, ...
}2 {(1)n+1}= {1,1,1,1,1, ...}
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TheoremIf lim
n |an | = 0, then limnan = 0.
NoteTotoo lang ito kapag zero ang limit.
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TheoremIf lim
n |an | = 0, then limnan = 0.
NoteTotoo lang ito kapag zero ang limit.
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Example
Use the theorem to show that the sequence{(1)n2n
}is convergent.
0
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Example
Use the theorem to show that the sequence{(1)n2n
}is convergent.
0
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RemarkLet {an} be an alternating sequence. If lim
n |an | 6= limn|an |, then limnandoes not exist. (That is, {an} is divergent.)
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ExampleUse the remark to show that the sequence
{(1)n+1} is divergent.
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ExampleUse the remark to show that the sequence
{(1)n+1} is divergent.
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Outline
1 Convergence of Alternating Sequences
2 Monotone and Bounded Sequences
3 Proving the convergence of sequences by definition
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DefinitionA sequence {an}n=1 is said to be
1 increasing if an an+1 for all n N2 decreasing if an an+1 for all n N3 monotonic if {an}n=1 is either increasing or decreasing.
Example
1
{1
n
}={1,1
2,1
3,1
4, ...
}is a decreasing sequence
2 {3n8}= {5,2,1,4, ..} is an increasing sequence.
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DefinitionA sequence {an}n=1 is said to be
1 increasing if
an an+1 for all n N2 decreasing if an an+1 for all n N3 monotonic if {an}n=1 is either increasing or decreasing.
Example
1
{1
n
}={1,1
2,1
3,1
4, ...
}is a decreasing sequence
2 {3n8}= {5,2,1,4, ..} is an increasing sequence.
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DefinitionA sequence {an}n=1 is said to be
1 increasing if an an+1 for all n N
2 decreasing if an an+1 for all n N3 monotonic if {an}n=1 is either increasing or decreasing.
Example
1
{1
n
}={1,1
2,1
3,1
4, ...
}is a decreasing sequence
2 {3n8}= {5,2,1,4, ..} is an increasing sequence.
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DefinitionA sequence {an}n=1 is said to be
1 increasing if an an+1 for all n N2 decreasing if
an an+1 for all n N3 monotonic if {an}n=1 is either increasing or decreasing.
Example
1
{1
n
}={1,1
2,1
3,1
4, ...
}is a decreasing sequence
2 {3n8}= {5,2,1,4, ..} is an increasing sequence.
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DefinitionA sequence {an}n=1 is said to be
1 increasing if an an+1 for all n N2 decreasing if an an+1 for all n N
3 monotonic if {an}n=1 is either increasing or decreasing.
Example
1
{1
n
}={1,1
2,1
3,1
4, ...
}is a decreasing sequence
2 {3n8}= {5,2,1,4, ..} is an increasing sequence.
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DefinitionA sequence {an}n=1 is said to be
1 increasing if an an+1 for all n N2 decreasing if an an+1 for all n N3 monotonic if
{an}n=1 is either increasing or decreasing.
Example
1
{1
n
}={1,1
2,1
3,1
4, ...
}is a decreasing sequence
2 {3n8}= {5,2,1,4, ..} is an increasing sequence.
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DefinitionA sequence {an}n=1 is said to be
1 increasing if an an+1 for all n N2 decreasing if an an+1 for all n N3 monotonic if {an}n=1 is either increasing or decreasing.
Example
1
{1
n
}={1,1
2,1
3,1
4, ...
}is a decreasing sequence
2 {3n8}= {5,2,1,4, ..} is an increasing sequence.
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DefinitionA sequence {an}n=1 is said to be
1 increasing if an an+1 for all n N2 decreasing if an an+1 for all n N3 monotonic if {an}n=1 is either increasing or decreasing.
Example
1
{1
n
}={1,1
2,1
3,1
4, ...
}is a decreasing sequence
2 {3n8}= {5,2,1,4, ..} is an increasing sequence.
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DefinitionA sequence {an}n=1 is said to be
1 increasing if an an+1 for all n N2 decreasing if an an+1 for all n N3 monotonic if {an}n=1 is either increasing or decreasing.
Example
1
{1
n
}={1,1
2,1
3,1
4, ...
}is a decreasing sequence
2 {3n8}= {5,2,1,4, ..} is an increasing sequence.
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RemarkTwo methods to show that a sequence {an} is monotonic:
1 Let f (n)= an . Consider f (n) as a function of R. Find f (n).increasing: f (n) 0 for all n [k,)decreasing: f (n) 0 for all n [k,)
2 If the terms of {an} are all positive:increasing:
an+1an
1 for all n Idecreasing:
an+1an
1 for all n I
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RemarkTwo methods to show that a sequence {an} is monotonic:
1 Let f (n)= an . Consider f (n) as a function of R. Find f (n).increasing: f (n) 0 for all n [k,)decreasing: f (n) 0 for all n [k,)
2 If the terms of {an} are all positive:increasing:
an+1an
1 for all n Idecreasing:
an+1an
1 for all n I
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ExampleDetermine whether or not the following sequences are monotonic:
1
{5n+12n
}
decreasing
2
{n!
10
}increasing
3
{1
(2n)!
}decreasing
4
{(1)n2n
}not monotonic
5
{cos
pi
n
}n=2
increasing
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ExampleDetermine whether or not the following sequences are monotonic:
1
{5n+12n
}decreasing
2
{n!
10
}increasing
3
{1
(2n)!
}decreasing
4
{(1)n2n
}not monotonic
5
{cos
pi
n
}n=2
increasing
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ExampleDetermine whether or not the following sequences are monotonic:
1
{5n+12n
}decreasing
2
{n!
10
}
increasing
3
{1
(2n)!
}decreasing
4
{(1)n2n
}not monotonic
5
{cos
pi
n
}n=2
increasing
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ExampleDetermine whether or not the following sequences are monotonic:
1
{5n+12n
}decreasing
2
{n!
10
}increasing
3
{1
(2n)!
}decreasing
4
{(1)n2n
}not monotonic
5
{cos
pi
n
}n=2
increasing
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ExampleDetermine whether or not the following sequences are monotonic:
1
{5n+12n
}decreasing
2
{n!
10
}increasing
3
{1
(2n)!
}
decreasing
4
{(1)n2n
}not monotonic
5
{cos
pi
n
}n=2
increasing
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ExampleDetermine whether or not the following sequences are monotonic:
1
{5n+12n
}decreasing
2
{n!
10
}increasing
3
{1
(2n)!
}decreasing
4
{(1)n2n
}not monotonic
5
{cos
pi
n
}n=2
increasing
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ExampleDetermine whether or not the following sequences are monotonic:
1
{5n+12n
}decreasing
2
{n!
10
}increasing
3
{1
(2n)!
}decreasing
4
{(1)n2n
}
not monotonic
5
{cos
pi
n
}n=2
increasing
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ExampleDetermine whether or not the following sequences are monotonic:
1
{5n+12n
}decreasing
2
{n!
10
}increasing
3
{1
(2n)!
}decreasing
4
{(1)n2n
}not monotonic
5
{cos
pi
n
}n=2
increasing
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ExampleDetermine whether or not the following sequences are monotonic:
1
{5n+12n
}decreasing
2
{n!
10
}increasing
3
{1
(2n)!
}decreasing
4
{(1)n2n
}not monotonic
5
{cos
pi
n
}n=2
increasing
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ExampleDetermine whether or not the following sequences are monotonic:
1
{5n+12n
}decreasing
2
{n!
10
}increasing
3
{1
(2n)!
}decreasing
4
{(1)n2n
}not monotonic
5
{cos
pi
n
}n=2
increasing
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DefinitionIf {an}n=1 is a sequence, then
1 l is a lower bound of the sequence if l an for all n N.2 u is an upper bound of the sequence if u an for all n N.3 The sequence is bounded if it has both an upper bound and lower
bound.
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DefinitionIf {an}n=1 is a sequence, then
1 l is a lower bound of the sequence if
l an for all n N.2 u is an upper bound of the sequence if u an for all n N.3 The sequence is bounded if it has both an upper bound and lower
bound.
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DefinitionIf {an}n=1 is a sequence, then
1 l is a lower bound of the sequence if l an for all n N.
2 u is an upper bound of the sequence if u an for all n N.3 The sequence is bounded if it has both an upper bound and lower
bound.
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DefinitionIf {an}n=1 is a sequence, then
1 l is a lower bound of the sequence if l an for all n N.2 u is an upper bound of the sequence if
u an for all n N.3 The sequence is bounded if it has both an upper bound and lower
bound.
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DefinitionIf {an}n=1 is a sequence, then
1 l is a lower bound of the sequence if l an for all n N.2 u is an upper bound of the sequence if u an for all n N.
3 The sequence is bounded if it has both an upper bound and lowerbound.
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DefinitionIf {an}n=1 is a sequence, then
1 l is a lower bound of the sequence if l an for all n N.2 u is an upper bound of the sequence if u an for all n N.3 The sequence is bounded if
it has both an upper bound and lowerbound.
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DefinitionIf {an}n=1 is a sequence, then
1 l is a lower bound of the sequence if l an for all n N.2 u is an upper bound of the sequence if u an for all n N.3 The sequence is bounded if it has both an upper bound and lower
bound.
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ExampleWhich among the sequences in the previous example are bounded?
1
{5n+12n
}(decreasing)
bounded
2
{n!
10
}(increasing) not bounded
3
{1
(2n)!
}(decreasing) bounded
4
{cos
pi
n
}n=2
(increasing) bounded
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ExampleWhich among the sequences in the previous example are bounded?
1
{5n+12n
}(decreasing) bounded
2
{n!
10
}(increasing) not bounded
3
{1
(2n)!
}(decreasing) bounded
4
{cos
pi
n
}n=2
(increasing) bounded
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ExampleWhich among the sequences in the previous example are bounded?
1
{5n+12n
}(decreasing) bounded
2
{n!
10
}(increasing)
not bounded
3
{1
(2n)!
}(decreasing) bounded
4
{cos
pi
n
}n=2
(increasing) bounded
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ExampleWhich among the sequences in the previous example are bounded?
1
{5n+12n
}(decreasing) bounded
2
{n!
10
}(increasing) not bounded
3
{1
(2n)!
}(decreasing) bounded
4
{cos
pi
n
}n=2
(increasing) bounded
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ExampleWhich among the sequences in the previous example are bounded?
1
{5n+12n
}(decreasing) bounded
2
{n!
10
}(increasing) not bounded
3
{1
(2n)!
}(decreasing)
bounded
4
{cos
pi
n
}n=2
(increasing) bounded
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ExampleWhich among the sequences in the previous example are bounded?
1
{5n+12n
}(decreasing) bounded
2
{n!
10
}(increasing) not bounded
3
{1
(2n)!
}(decreasing) bounded
4
{cos
pi
n
}n=2
(increasing) bounded
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ExampleWhich among the sequences in the previous example are bounded?
1
{5n+12n
}(decreasing) bounded
2
{n!
10
}(increasing) not bounded
3
{1
(2n)!
}(decreasing) bounded
4
{cos
pi
n
}n=2
(increasing)
bounded
-
ExampleWhich among the sequences in the previous example are bounded?
1
{5n+12n
}(decreasing) bounded
2
{n!
10
}(increasing) not bounded
3
{1
(2n)!
}(decreasing) bounded
4
{cos
pi
n
}n=2
(increasing) bounded
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Is the sequence{(1)n2n
}bounded? (not monotonic)
0
BOUNDED!
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Is the sequence{(1)n2n
}bounded? (not monotonic)
0
BOUNDED!
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Is the sequence{(1)n2n
}bounded? (not monotonic)
0
BOUNDED!
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Theorem (Monotone Convergence Theorem)A bounded monotonic sequence is convergent.
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ExampleWhich among the sequences in the previous examples are convergent bythe theorem?
1
{5n+12n
}(decreasing, bounded)
convergent
2
{n!
10
}(increasing, not bounded) no conclusion
3
{1
(2n)!
}(decreasing, bounded) convergent
4
{(1)n2n
}(not monotonic, bounded) no conclusion
5
{cos
pi
n
}n=2
(increasing, bounded) convergent
-
ExampleWhich among the sequences in the previous examples are convergent bythe theorem?
1
{5n+12n
}(decreasing, bounded) convergent
2
{n!
10
}(increasing, not bounded) no conclusion
3
{1
(2n)!
}(decreasing, bounded) convergent
4
{(1)n2n
}(not monotonic, bounded) no conclusion
5
{cos
pi
n
}n=2
(increasing, bounded) convergent
-
ExampleWhich among the sequences in the previous examples are convergent bythe theorem?
1
{5n+12n
}(decreasing, bounded) convergent
2
{n!
10
}(increasing, not bounded)
no conclusion
3
{1
(2n)!
}(decreasing, bounded) convergent
4
{(1)n2n
}(not monotonic, bounded) no conclusion
5
{cos
pi
n
}n=2
(increasing, bounded) convergent
-
ExampleWhich among the sequences in the previous examples are convergent bythe theorem?
1
{5n+12n
}(decreasing, bounded) convergent
2
{n!
10
}(increasing, not bounded) no conclusion
3
{1
(2n)!
}(decreasing, bounded) convergent
4
{(1)n2n
}(not monotonic, bounded) no conclusion
5
{cos
pi
n
}n=2
(increasing, bounded) convergent
-
ExampleWhich among the sequences in the previous examples are convergent bythe theorem?
1
{5n+12n
}(decreasing, bounded) convergent
2
{n!
10
}(increasing, not bounded) no conclusion
3
{1
(2n)!
}(decreasing, bounded)
convergent
4
{(1)n2n
}(not monotonic, bounded) no conclusion
5
{cos
pi
n
}n=2
(increasing, bounded) convergent
-
ExampleWhich among the sequences in the previous examples are convergent bythe theorem?
1
{5n+12n
}(decreasing, bounded) convergent
2
{n!
10
}(increasing, not bounded) no conclusion
3
{1
(2n)!
}(decreasing, bounded) convergent
4
{(1)n2n
}(not monotonic, bounded) no conclusion
5
{cos
pi
n
}n=2
(increasing, bounded) convergent
-
ExampleWhich among the sequences in the previous examples are convergent bythe theorem?
1
{5n+12n
}(decreasing, bounded) convergent
2
{n!
10
}(increasing, not bounded) no conclusion
3
{1
(2n)!
}(decreasing, bounded) convergent
4
{(1)n2n
}(not monotonic, bounded)
no conclusion
5
{cos
pi
n
}n=2
(increasing, bounded) convergent
-
ExampleWhich among the sequences in the previous examples are convergent bythe theorem?
1
{5n+12n
}(decreasing, bounded) convergent
2
{n!
10
}(increasing, not bounded) no conclusion
3
{1
(2n)!
}(decreasing, bounded) convergent
4
{(1)n2n
}(not monotonic, bounded) no conclusion
5
{cos
pi
n
}n=2
(increasing, bounded) convergent
-
ExampleWhich among the sequences in the previous examples are convergent bythe theorem?
1
{5n+12n
}(decreasing, bounded) convergent
2
{n!
10
}(increasing, not bounded) no conclusion
3
{1
(2n)!
}(decreasing, bounded) convergent
4
{(1)n2n
}(not monotonic, bounded) no conclusion
5
{cos
pi
n
}n=2
(increasing, bounded)
convergent
-
ExampleWhich among the sequences in the previous examples are convergent bythe theorem?
1
{5n+12n
}(decreasing, bounded) convergent
2
{n!
10
}(increasing, not bounded) no conclusion
3
{1
(2n)!
}(decreasing, bounded) convergent
4
{(1)n2n
}(not monotonic, bounded) no conclusion
5
{cos
pi
n
}n=2
(increasing, bounded) convergent
-
ExampleWhich among the sequences in the previous examples are convergent bythe theorem?
1
{5n+12n
}(decreasing, bounded) convergent
2
{n!
10
}(increasing, not bounded) no conclusion
3
{1
(2n)!
}(decreasing, bounded) convergent
4
{(1)n2n
}(not monotonic, bounded) no conclusion
5
{cos
pi
n
}n=2
(increasing, bounded) convergent
-
Completeness AxiomLet S be a nonempty set S of real numbers. If S has an upperbound, thenS has a smallest upper bound (called the least upper bound). If S has alower bound, then S has largest lower bound (called the greatest lowerbound).
-
RemarkLet {an} be a bounded monotonic sequence.
1 increasing: glb = first term, lub = limit of the sequence2 decreasing: lub = first term, glb = limit of the sequence
ExampleDetermine the glb and lub of the convergent sequences in the previousexample.
1
{5n+12n
}(decreasing)
2
{1
(2n)!
}(decreasing)
3
{cos
pi
n
}n=2
(increasing)
-
RemarkLet {an} be a bounded monotonic sequence.
1 increasing: glb =
first term, lub = limit of the sequence2 decreasing: lub = first term, glb = limit of the sequence
ExampleDetermine the glb and lub of the convergent sequences in the previousexample.
1
{5n+12n
}(decreasing)
2
{1
(2n)!
}(decreasing)
3
{cos
pi
n
}n=2
(increasing)
-
RemarkLet {an} be a bounded monotonic sequence.
1 increasing: glb = first term,
lub = limit of the sequence2 decreasing: lub = first term, glb = limit of the sequence
ExampleDetermine the glb and lub of the convergent sequences in the previousexample.
1
{5n+12n
}(decreasing)
2
{1
(2n)!
}(decreasing)
3
{cos
pi
n
}n=2
(increasing)
-
RemarkLet {an} be a bounded monotonic sequence.
1 increasing: glb = first term, lub = limit of the sequence
2 decreasing: lub = first term, glb = limit of the sequence
ExampleDetermine the glb and lub of the convergent sequences in the previousexample.
1
{5n+12n
}(decreasing)
2
{1
(2n)!
}(decreasing)
3
{cos
pi
n
}n=2
(increasing)
-
RemarkLet {an} be a bounded monotonic sequence.
1 increasing: glb = first term, lub = limit of the sequence2 decreasing: lub =
first term, glb = limit of the sequence
ExampleDetermine the glb and lub of the convergent sequences in the previousexample.
1
{5n+12n
}(decreasing)
2
{1
(2n)!
}(decreasing)
3
{cos
pi
n
}n=2
(increasing)
-
RemarkLet {an} be a bounded monotonic sequence.
1 increasing: glb = first term, lub = limit of the sequence2 decreasing: lub = first term,
glb = limit of the sequence
ExampleDetermine the glb and lub of the convergent sequences in the previousexample.
1
{5n+12n
}(decreasing)
2
{1
(2n)!
}(decreasing)
3
{cos
pi
n
}n=2
(increasing)
-
RemarkLet {an} be a bounded monotonic sequence.
1 increasing: glb = first term, lub = limit of the sequence2 decreasing: lub = first term, glb = limit of the sequence
ExampleDetermine the glb and lub of the convergent sequences in the previousexample.
1
{5n+12n
}(decreasing)
2
{1
(2n)!
}(decreasing)
3
{cos
pi
n
}n=2
(increasing)
-
RemarkLet {an} be a bounded monotonic sequence.
1 increasing: glb = first term, lub = limit of the sequence2 decreasing: lub = first term, glb = limit of the sequence
ExampleDetermine the glb and lub of the convergent sequences in the previousexample.
1
{5n+12n
}(decreasing)
2
{1
(2n)!
}(decreasing)
3
{cos
pi
n
}n=2
(increasing)
-
RemarkLet {an} be a bounded monotonic sequence.
1 increasing: glb = first term, lub = limit of the sequence2 decreasing: lub = first term, glb = limit of the sequence
ExampleDetermine the glb and lub of the convergent sequences in the previousexample.
1
{5n+12n
}(decreasing)
2
{1
(2n)!
}(decreasing)
3
{cos
pi
n
}n=2
(increasing)
-
RemarkLet {an} be a bounded monotonic sequence.
1 increasing: glb = first term, lub = limit of the sequence2 decreasing: lub = first term, glb = limit of the sequence
ExampleDetermine the glb and lub of the convergent sequences in the previousexample.
1
{5n+12n
}(decreasing)
2
{1
(2n)!
}(decreasing)
3
{cos
pi
n
}n=2
(increasing)
-
RemarkLet {an} be a bounded monotonic sequence.
1 increasing: glb = first term, lub = limit of the sequence2 decreasing: lub = first term, glb = limit of the sequence
ExampleDetermine the glb and lub of the convergent sequences in the previousexample.
1
{5n+12n
}(decreasing)
2
{1
(2n)!
}(decreasing)
3
{cos
pi
n
}n=2
(increasing)
-
Outline
1 Convergence of Alternating Sequences
2 Monotone and Bounded Sequences
3 Proving the convergence of sequences by definition
-
Please note some changes...
Definition (Convergence of a Sequence)A sequence {an} is said to converge to the limit L if
given any > 0, thereexists a number N such that:
If n is an integer and n >N , then |an L| < .
-
Please note some changes...
Definition (Convergence of a Sequence)A sequence {an} is said to converge to the limit L if given any > 0,
thereexists a number N such that:
If n is an integer and n >N , then |an L| < .
-
Please note some changes...
Definition (Convergence of a Sequence)A sequence {an} is said to converge to the limit L if given any > 0, thereexists a number N such that:
If n is an integer and n >N , then |an L| < .
-
Please note some changes...
Definition (Convergence of a Sequence)A sequence {an} is said to converge to the limit L if given any > 0, thereexists a number N such that:
If n is an integer and n >N , then |an L| < .
-
0L
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0L
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0L
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N
LL+
N
-
0L
L
L+
N
L
L+
N
LL+
N
LL+
N
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RemarkTo show that {an} converges to L by definition:
1 Find a suitable N in terms of 2 Show that the given N satisfies the condition: If n >N , then|an L| < .
-
RemarkTo show that {an} converges to L by definition:
1 Find a suitable N in terms of
2 Show that the given N satisfies the condition: If n >N , then|an L| < .
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RemarkTo show that {an} converges to L by definition:
1 Find a suitable N in terms of 2 Show that the given N satisfies the condition: If n >N , then|an L| < .
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Tip:1 Simplify |an L|
2 In the inequality |an L| < , solve for n in terms of . The theexpression in terms of is your candidate for N .
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Tip:1 Simplify |an L|2 In the inequality |an L| < , solve for n in terms of . The the
expression in terms of is your candidate for N .
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ExampleProve the following using the definition of convergence of a sequence:
1{ 1n
}converges to 0.
2{2n1
5n
}converges to 25 .
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Announcements
1 Do not forget to bring your SW blue book to the next class!
Convergence of Alternating SequencesMonotone and Bounded SequencesProving the convergence of sequences by definition
