MATH 38Mathematical Analysis III
I. F. Evidente
IMSP (UPLB)
Outline
1 Sequences
2 Graphs of Sequences
3 Convergence of a Sequence
4 Squeeze Theorem for Sequences
Recall:
Some limit forms (Not indeterminate!)
(e): limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Recall:
Some limit forms (Not indeterminate!)(e):
limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Recall:
Some limit forms (Not indeterminate!)(e): limit is
(e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Recall:
Some limit forms (Not indeterminate!)(e): limit is (e):
limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Recall:
Some limit forms (Not indeterminate!)(e): limit is (e): limit is 0
(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Recall:
Some limit forms (Not indeterminate!)(e): limit is (e): limit is 0(ln()):
limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Recall:
Some limit forms (Not indeterminate!)(e): limit is (e): limit is 0(ln()): limit is
(ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Recall:
Some limit forms (Not indeterminate!)(e): limit is (e): limit is 0(ln()): limit is (ln(0+)):
limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Recall:
Some limit forms (Not indeterminate!)(e): limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is
(Arctan()): limit is pi2(Arctan()): limit is pi2
Recall:
Some limit forms (Not indeterminate!)(e): limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()):
limit is pi2(Arctan()): limit is pi2
Recall:
Some limit forms (Not indeterminate!)(e): limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2
(Arctan()): limit is pi2
Recall:
Some limit forms (Not indeterminate!)(e): limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()):
limit is pi2
Recall:
Some limit forms (Not indeterminate!)(e): limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Examples
1 limx ln
(x2+12x1
)=
2 limx ln
(1
x
)=
3 limx0+
Arctan(lnx)= pi2
Examples
1 limx ln
(x2+12x1
)=
2 limx ln
(1
x
)=
3 limx0+
Arctan(lnx)= pi2
Examples
1 limx ln
(x2+12x1
)=
2 limx ln
(1
x
)=
3 limx0+
Arctan(lnx)= pi2
Examples
1 limx ln
(x2+12x1
)=
2 limx ln
(1
x
)=
3 limx0+
Arctan(lnx)= pi2
Examples
1 limx ln
(x2+12x1
)=
2 limx ln
(1
x
)=
3 limx0+
Arctan(lnx)=
pi2
Examples
1 limx ln
(x2+12x1
)=
2 limx ln
(1
x
)=
3 limx0+
Arctan(lnx)= pi2
Outline
1 Sequences
2 Graphs of Sequences
3 Convergence of a Sequence
4 Squeeze Theorem for Sequences
Sequences
What is a sequence?
A sequence is a succession of numbers whose order is determined bya rule.
Sequences
What is a sequence?A sequence is a succession of numbers whose order is determined bya rule.
What is the rule for the following sequences?
1 {5,2,1,4,7, ...}2{12 , 14 ,18 , 116 , ...}
3 {1,0,1,0, ...}
4 {1, 12 ,13 ,
14 , ...}
5 {1,12 , 13 ,14 , ...}6 {12 ,
23 ,
34 ,
45 , ...}
7 {0,1,1,2,3,5, ...}
What is the rule for the following sequences?1 {5,2,1,4,7, ...}
2{12 , 14 ,18 , 116 , ...}
3 {1,0,1,0, ...}
4 {1, 12 ,13 ,
14 , ...}
5 {1,12 , 13 ,14 , ...}6 {12 ,
23 ,
34 ,
45 , ...}
7 {0,1,1,2,3,5, ...}
What is the rule for the following sequences?1 {5,2,1,4,7, ...}2{12 , 14 ,18 , 116 , ...}
3 {1,0,1,0, ...}
4 {1, 12 ,13 ,
14 , ...}
5 {1,12 , 13 ,14 , ...}6 {12 ,
23 ,
34 ,
45 , ...}
7 {0,1,1,2,3,5, ...}
What is the rule for the following sequences?1 {5,2,1,4,7, ...}2{12 , 14 ,18 , 116 , ...}
3 {1,0,1,0, ...}
4 {1, 12 ,13 ,
14 , ...}
5 {1,12 , 13 ,14 , ...}6 {12 ,
23 ,
34 ,
45 , ...}
7 {0,1,1,2,3,5, ...}
What is the rule for the following sequences?1 {5,2,1,4,7, ...}2{12 , 14 ,18 , 116 , ...}
3 {1,0,1,0, ...}
4 {1, 12 ,13 ,
14 , ...}
5 {1,12 , 13 ,14 , ...}6 {12 ,
23 ,
34 ,
45 , ...}
7 {0,1,1,2,3,5, ...}
What is the rule for the following sequences?1 {5,2,1,4,7, ...}2{12 , 14 ,18 , 116 , ...}
3 {1,0,1,0, ...}
4 {1, 12 ,13 ,
14 , ...}
5 {1,12 , 13 ,14 , ...}
6 {12 ,23 ,
34 ,
45 , ...}
7 {0,1,1,2,3,5, ...}
What is the rule for the following sequences?1 {5,2,1,4,7, ...}2{12 , 14 ,18 , 116 , ...}
3 {1,0,1,0, ...}
4 {1, 12 ,13 ,
14 , ...}
5 {1,12 , 13 ,14 , ...}6 {12 ,
23 ,
34 ,
45 , ...}
7 {0,1,1,2,3,5, ...}
What is the rule for the following sequences?1 {5,2,1,4,7, ...}2{12 , 14 ,18 , 116 , ...}
3 {1,0,1,0, ...}
4 {1, 12 ,13 ,
14 , ...}
5 {1,12 , 13 ,14 , ...}6 {12 ,
23 ,
34 ,
45 , ...}
7 {0,1,1,2,3,5, ...}
Sequences
Focus: sequences with infinite terms, rule is defined by a function.
DefinitionA sequence of real numbers is the ordered range of some function f (n)whose domain I is an ordered set of the form {k,k+1,k+2, ...}, where k Z.
ExampleConsider f :NR, where f (n)=pn. What is the sequence arising fromthis function?
DefinitionA sequence of real numbers is the ordered range of some function f (n)whose domain I is an ordered set of the form {k,k+1,k+2, ...}, where k Z.
ExampleConsider f :NR, where f (n)=pn. What is the sequence arising fromthis function?
DefinitionA sequence of real numbers is the ordered range of some function f (n)whose domain I is an ordered set of the form {k,k+1,k+2, ...}, where k Z.
Terminologies and Notations:Indexing Set: I = {k,k+1,k+2, ...}, index, lower or starting index,usual lower index: 0,1,2Rule: f (n)Notation: {an}n=k , where an = f (n)If there is no ambiguity, we simply write {an}
DefinitionA sequence of real numbers is the ordered range of some function f (n)whose domain I is an ordered set of the form {k,k+1,k+2, ...}, where k Z.
Terminologies and Notations:Indexing Set: I = {k,k+1,k+2, ...},
index, lower or starting index,usual lower index: 0,1,2Rule: f (n)Notation: {an}n=k , where an = f (n)If there is no ambiguity, we simply write {an}
DefinitionA sequence of real numbers is the ordered range of some function f (n)whose domain I is an ordered set of the form {k,k+1,k+2, ...}, where k Z.
Terminologies and Notations:Indexing Set: I = {k,k+1,k+2, ...}, index,
lower or starting index,usual lower index: 0,1,2Rule: f (n)Notation: {an}n=k , where an = f (n)If there is no ambiguity, we simply write {an}
DefinitionA sequence of real numbers is the ordered range of some function f (n)whose domain I is an ordered set of the form {k,k+1,k+2, ...}, where k Z.
Terminologies and Notations:Indexing Set: I = {k,k+1,k+2, ...}, index, lower or starting index,
usual lower index: 0,1,2Rule: f (n)Notation: {an}n=k , where an = f (n)If there is no ambiguity, we simply write {an}
DefinitionA sequence of real numbers is the ordered range of some function f (n)whose domain I is an ordered set of the form {k,k+1,k+2, ...}, where k Z.
Terminologies and Notations:Indexing Set: I = {k,k+1,k+2, ...}, index, lower or starting index,usual lower index: 0,1,2
Rule: f (n)Notation: {an}n=k , where an = f (n)If there is no ambiguity, we simply write {an}
DefinitionA sequence of real numbers is the ordered range of some function f (n)whose domain I is an ordered set of the form {k,k+1,k+2, ...}, where k Z.
Terminologies and Notations:Indexing Set: I = {k,k+1,k+2, ...}, index, lower or starting index,usual lower index: 0,1,2Rule: f (n)
Notation: {an}n=k , where an = f (n)If there is no ambiguity, we simply write {an}
DefinitionA sequence of real numbers is the ordered range of some function f (n)whose domain I is an ordered set of the form {k,k+1,k+2, ...}, where k Z.
Terminologies and Notations:Indexing Set: I = {k,k+1,k+2, ...}, index, lower or starting index,usual lower index: 0,1,2Rule: f (n)Notation: {an}n=k , where an = f (n)
If there is no ambiguity, we simply write {an}
DefinitionA sequence of real numbers is the ordered range of some function f (n)whose domain I is an ordered set of the form {k,k+1,k+2, ...}, where k Z.
Terminologies and Notations:Indexing Set: I = {k,k+1,k+2, ...}, index, lower or starting index,usual lower index: 0,1,2Rule: f (n)Notation: {an}n=k , where an = f (n)If there is no ambiguity, we simply write {an}
What is the function defining the rule for the following sequences?1 {5,2,1,4,7, ...}=
{3n8}n=1
2{12 , 14 ,18 , 116 , ...}= {
(1)n2n
}n=1
3 {1,0,1,0, ...}
{an}n=1 where an =
{1 if n is even0 if n is odd
What is the function defining the rule for the following sequences?1 {5,2,1,4,7, ...}=
{3n8}n=1
2{12 , 14 ,18 , 116 , ...}= {
(1)n2n
}n=1
3 {1,0,1,0, ...}
{an}n=1 where an =
{1 if n is even0 if n is odd
What is the function defining the rule for the following sequences?1 {5,2,1,4,7, ...}=
{3n8}n=1
2{12 , 14 ,18 , 116 , ...}=
{(1)n2n
}n=1
3 {1,0,1,0, ...}
{an}n=1 where an =
{1 if n is even0 if n is odd
What is the function defining the rule for the following sequences?1 {5,2,1,4,7, ...}=
{3n8}n=1
2{12 , 14 ,18 , 116 , ...}= {
(1)n2n
}n=1
3 {1,0,1,0, ...}
{an}n=1 where an =
{1 if n is even0 if n is odd
What is the function defining the rule for the following sequences?1 {5,2,1,4,7, ...}=
{3n8}n=1
2{12 , 14 ,18 , 116 , ...}= {
(1)n2n
}n=1
3 {1,0,1,0, ...}
{an}n=1 where an =
{1 if n is even0 if n is odd
What is the function defining the rule for the following sequences?1 {5,2,1,4,7, ...}=
{3n8}n=1
2{12 , 14 ,18 , 116 , ...}= {
(1)n2n
}n=1
3 {1,0,1,0, ...}
{an}n=1 where an =
{1 if n is even0 if n is odd
What is the function defining the rule for the following sequences?1 {1, 12 ,
13 ,
14 , ...}=
{1
n
}n=1
2 {1,12 , 13 ,14 , ...}= {(1)n+1
n
}n=1
3 {12 ,23 ,
34 ,
45 , ...}= { n
n+1}
What is the function defining the rule for the following sequences?1 {1, 12 ,
13 ,
14 , ...}= {
1
n
}n=1
2 {1,12 , 13 ,14 , ...}= {(1)n+1
n
}n=1
3 {12 ,23 ,
34 ,
45 , ...}= { n
n+1}
What is the function defining the rule for the following sequences?1 {1, 12 ,
13 ,
14 , ...}= {
1
n
}n=1
2 {1,12 , 13 ,14 , ...}=
{(1)n+1
n
}n=1
3 {12 ,23 ,
34 ,
45 , ...}= { n
n+1}
What is the function defining the rule for the following sequences?1 {1, 12 ,
13 ,
14 , ...}= {
1
n
}n=1
2 {1,12 , 13 ,14 , ...}= {(1)n+1
n
}n=1
3 {12 ,23 ,
34 ,
45 , ...}= { n
n+1}
What is the function defining the rule for the following sequences?1 {1, 12 ,
13 ,
14 , ...}= {
1
n
}n=1
2 {1,12 , 13 ,14 , ...}= {(1)n+1
n
}n=1
3 {12 ,23 ,
34 ,
45 , ...}=
{ nn+1
}
What is the function defining the rule for the following sequences?1 {1, 12 ,
13 ,
14 , ...}= {
1
n
}n=1
2 {1,12 , 13 ,14 , ...}= {(1)n+1
n
}n=1
3 {12 ,23 ,
34 ,
45 , ...}= { n
n+1}
Outline
1 Sequences
2 Graphs of Sequences
3 Convergence of a Sequence
4 Squeeze Theorem for Sequences
Graph of a SequenceThe graph of a sequence {an} is the geometric representation on theCartesian plane of the the set
{(n,an) | n I }
Example{3n8}n=1
0
Example{3n8}n=1
0
Example{(1)n2n
}n=1
0
Example{(1)n2n
}n=1
0
Example{1,0,1,0, ...}
0
Example{1,0,1,0, ...}
0
Example{ 1n
}n=1
0
Example{ 1n
}n=1
0
Example{(1)n+1
n
}n=1
0
Example{(1)n+1
n
}n=1
0
Example{ nn+1}n=1
0
Example{ nn+1}n=1
0
Outline
1 Sequences
2 Graphs of Sequences
3 Convergence of a Sequence
4 Squeeze Theorem for Sequences
QuestionWhat is the behavior of a sequence at the tail end, when n is very, verylarge?
(Mathematicians say: for all values of n greater than some fixed index N .)
QuestionWhat is the behavior of a sequence at the tail end, when n is very, verylarge?(Mathematicians say: for all values of n greater than some fixed index N .)
Definition (Convergence of a Sequence)A sequence {an} is said to converge to the limit L if
given any > 0, thereexists a positive integer N such that
|an L| < for all n N .
A sequence that does not converge is said to diverge.
Notation:limnan = L, or an L as n.
Definition (Convergence of a Sequence)A sequence {an} is said to converge to the limit L if given any > 0,
thereexists a positive integer N such that
|an L| < for all n N .
A sequence that does not converge is said to diverge.
Notation:limnan = L, or an L as n.
Definition (Convergence of a Sequence)A sequence {an} is said to converge to the limit L if given any > 0, thereexists a positive integer N such that
|an L| < for all n N .
A sequence that does not converge is said to diverge.
Notation:limnan = L, or an L as n.
Definition (Convergence of a Sequence)A sequence {an} is said to converge to the limit L if given any > 0, thereexists a positive integer N such that
|an L| < for all n N .
A sequence that does not converge is said to diverge.
Notation:limnan = L, or an L as n.
Definition (Convergence of a Sequence)A sequence {an} is said to converge to the limit L if given any > 0, thereexists a positive integer N such that
|an L| < for all n N .
A sequence that does not converge is said to diverge.
Notation:limnan = L, or an L as n.
Definition (Convergence of a Sequence)A sequence {an} is said to converge to the limit L if given any > 0, thereexists a positive integer N such that
|an L| < for all n N .
A sequence that does not converge is said to diverge.
Notation:limnan = L, or an L as n.
RemarkIntuitively, the limit of {an} is the unique value L that an approachesas n.
For a sequence to converge, L must be a real number (and cannot be).
RemarkIntuitively, the limit of {an} is the unique value L that an approachesas n.For a sequence to converge, L must be a real number (and cannot be).
Example
1 {3n8}n=1
is divergent.
2
{(1)n2n
}n=1 converges to 0.
3 {1,0,1,0, ...} is divergent.4{ 1n
}n=1 converges to 0.
5
{(1)n+1
n
}n=1 converges to 0.
6{ nn+1}n=1 converges to 1.
Example
1 {3n8}n=1 is divergent.
2
{(1)n2n
}n=1 converges to 0.
3 {1,0,1,0, ...} is divergent.4{ 1n
}n=1 converges to 0.
5
{(1)n+1
n
}n=1 converges to 0.
6{ nn+1}n=1 converges to 1.
Example
1 {3n8}n=1 is divergent.2
{(1)n2n
}n=1
converges to 0.
3 {1,0,1,0, ...} is divergent.4{ 1n
}n=1 converges to 0.
5
{(1)n+1
n
}n=1 converges to 0.
6{ nn+1}n=1 converges to 1.
Example
1 {3n8}n=1 is divergent.2
{(1)n2n
}n=1 converges to 0.
3 {1,0,1,0, ...} is divergent.4{ 1n
}n=1 converges to 0.
5
{(1)n+1
n
}n=1 converges to 0.
6{ nn+1}n=1 converges to 1.
Example
1 {3n8}n=1 is divergent.2
{(1)n2n
}n=1 converges to 0.
3 {1,0,1,0, ...}
is divergent.4{ 1n
}n=1 converges to 0.
5
{(1)n+1
n
}n=1 converges to 0.
6{ nn+1}n=1 converges to 1.
Example
1 {3n8}n=1 is divergent.2
{(1)n2n
}n=1 converges to 0.
3 {1,0,1,0, ...} is divergent.
4{ 1n
}n=1 converges to 0.
5
{(1)n+1
n
}n=1 converges to 0.
6{ nn+1}n=1 converges to 1.
Example
1 {3n8}n=1 is divergent.2
{(1)n2n
}n=1 converges to 0.
3 {1,0,1,0, ...} is divergent.4{ 1n
}n=1
converges to 0.
5
{(1)n+1
n
}n=1 converges to 0.
6{ nn+1}n=1 converges to 1.
Example
1 {3n8}n=1 is divergent.2
{(1)n2n
}n=1 converges to 0.
3 {1,0,1,0, ...} is divergent.4{ 1n
}n=1 converges to 0.
5
{(1)n+1
n
}n=1 converges to 0.
6{ nn+1}n=1 converges to 1.
Example
1 {3n8}n=1 is divergent.2
{(1)n2n
}n=1 converges to 0.
3 {1,0,1,0, ...} is divergent.4{ 1n
}n=1 converges to 0.
5
{(1)n+1
n
}n=1
converges to 0.
6{ nn+1}n=1 converges to 1.
Example
1 {3n8}n=1 is divergent.2
{(1)n2n
}n=1 converges to 0.
3 {1,0,1,0, ...} is divergent.4{ 1n
}n=1 converges to 0.
5
{(1)n+1
n
}n=1 converges to 0.
6{ nn+1}n=1 converges to 1.
Example
1 {3n8}n=1 is divergent.2
{(1)n2n
}n=1 converges to 0.
3 {1,0,1,0, ...} is divergent.4{ 1n
}n=1 converges to 0.
5
{(1)n+1
n
}n=1 converges to 0.
6{ nn+1}n=1
converges to 1.
Example
1 {3n8}n=1 is divergent.2
{(1)n2n
}n=1 converges to 0.
3 {1,0,1,0, ...} is divergent.4{ 1n
}n=1 converges to 0.
5
{(1)n+1
n
}n=1 converges to 0.
6{ nn+1}n=1 converges to 1.
Consider the behavior of{ 1n
}, where f (n)= 1n
Consider the behavior of function f :R\ {0}R\ {0}, where f (x)= 1x.
0
Consider the behavior of{ 1n
}, where f (n)= 1n
Consider the behavior of function f :R\ {0}R\ {0}, where f (x)= 1x.
0
Consider the behavior of{ 1n
}, where f (n)= 1n
Consider the behavior of function f :R\ {0}R\ {0}, where f (x)= 1x.
0
RemarkLet {an} be a sequence, where an = f (n).
Consider f (x), f :RR. Thenthe behavior of f (n) as n "follows" the behavior of f (x) as x.
RemarkLet {an} be a sequence, where an = f (n). Consider f (x), f :RR.
Thenthe behavior of f (n) as n "follows" the behavior of f (x) as x.
RemarkLet {an} be a sequence, where an = f (n). Consider f (x), f :RR. Thenthe behavior of f (n) as n "follows" the behavior of f (x) as x.
TheoremLet {an}n=1, where an = f (n). Consider f (x), f :RR. If limx f (x)= L(or ), then lim
n f (n)= L (or ).
If as x, f (x) L, or , then
f (n) will "follow" the behavior off (x).But not the other way around!Consider {sinpin} and the function f (x)= sinpix.
0
If as x, f (x) L, or , then f (n) will "follow" the behavior off (x).
But not the other way around!Consider {sinpin} and the function f (x)= sinpix.
0
If as x, f (x) L, or , then f (n) will "follow" the behavior off (x).But not the other way around!
Consider {sinpin} and the function f (x)= sinpix.
0
If as x, f (x) L, or , then f (n) will "follow" the behavior off (x).But not the other way around!Consider {sinpin} and the function f (x)= sinpix.
0
If as x, f (x) L, or , then f (n) will "follow" the behavior off (x).But not the other way around!Consider {sinpin} and the function f (x)= sinpix.
0
If as x, f (x) L, or , then f (n) will "follow" the behavior off (x).But not the other way around!Consider {sinpin} and the function f (x)= sinpix.
0
If as x, f (x) L, or , then f (n) will "follow" the behavior off (x).But not the other way around!Consider {sinpin} and the function f (x)= sinpix.
0
TheoremLet {an}n=1, where an = f (n). Consider f (x), f :RR. If limx f (x)= L(or ), then lim
n f (n)= L (or ).
Usefulness of theorem:To compute lim
n f (n), we can use limit theorems for function f :RR(thms from Math 36 and 37!)
TheoremLet {an}n=1, where an = f (n). Consider f (x), f :RR. If limx f (x)= L(or ), then lim
n f (n)= L (or ).
Usefulness of theorem:
To compute limn f (n), we can use limit theorems for function f :RR
(thms from Math 36 and 37!)
TheoremLet {an}n=1, where an = f (n). Consider f (x), f :RR. If limx f (x)= L(or ), then lim
n f (n)= L (or ).
Usefulness of theorem:To compute lim
n f (n), we can use limit theorems for function f :RR(thms from Math 36 and 37!)
Example
1 {3n8}n=1
limn3n8=
2{ 1n
}n=1
limn
1
n= 0
3{ nn+1}
limn
n
n+1 = 1
Example
1 {3n8}n=1limn3n8=
2{ 1n
}n=1
limn
1
n= 0
3{ nn+1}
limn
n
n+1 = 1
Example
1 {3n8}n=1limn3n8=
2{ 1n
}n=1
limn
1
n= 0
3{ nn+1}
limn
n
n+1 = 1
Example
1 {3n8}n=1limn3n8=
2{ 1n
}n=1
limn
1
n= 0
3{ nn+1}
limn
n
n+1 = 1
Example
1 {3n8}n=1limn3n8=
2{ 1n
}n=1
limn
1
n=
0
3{ nn+1}
limn
n
n+1 = 1
Example
1 {3n8}n=1limn3n8=
2{ 1n
}n=1
limn
1
n= 0
3{ nn+1}
limn
n
n+1 = 1
Example
1 {3n8}n=1limn3n8=
2{ 1n
}n=1
limn
1
n= 0
3{ nn+1}
limn
n
n+1 = 1
Example
1 {3n8}n=1limn3n8=
2{ 1n
}n=1
limn
1
n= 0
3{ nn+1}
limn
n
n+1 =
1
Example
1 {3n8}n=1limn3n8=
2{ 1n
}n=1
limn
1
n= 0
3{ nn+1}
limn
n
n+1 = 1
ExampleDetermine whether the following sequences converge or diverge.
1
{n3+n2+2n+13n3+n+5
}2 {nn2}3
{ln
(1
n1)}
n=2
4
{tan1n 1
n
}5
{2n
1+2n}
6
{( nn+1
)n}7
{1
n!
}
ExampleDetermine whether the following sequences converge or diverge.
1
{n3+n2+2n+13n3+n+5
}
2 {nn2}3
{ln
(1
n1)}
n=2
4
{tan1n 1
n
}5
{2n
1+2n}
6
{( nn+1
)n}7
{1
n!
}
ExampleDetermine whether the following sequences converge or diverge.
1
{n3+n2+2n+13n3+n+5
}2 {nn2}
3
{ln
(1
n1)}
n=2
4
{tan1n 1
n
}5
{2n
1+2n}
6
{( nn+1
)n}7
{1
n!
}
ExampleDetermine whether the following sequences converge or diverge.
1
{n3+n2+2n+13n3+n+5
}2 {nn2}3
{ln
(1
n1)}
n=2
4
{tan1n 1
n
}5
{2n
1+2n}
6
{( nn+1
)n}7
{1
n!
}
ExampleDetermine whether the following sequences converge or diverge.
1
{n3+n2+2n+13n3+n+5
}2 {nn2}3
{ln
(1
n1)}
n=2
4
{tan1n 1
n
}
5
{2n
1+2n}
6
{( nn+1
)n}7
{1
n!
}
ExampleDetermine whether the following sequences converge or diverge.
1
{n3+n2+2n+13n3+n+5
}2 {nn2}3
{ln
(1
n1)}
n=2
4
{tan1n 1
n
}5
{2n
1+2n}
6
{( nn+1
)n}7
{1
n!
}
ExampleDetermine whether the following sequences converge or diverge.
1
{n3+n2+2n+13n3+n+5
}2 {nn2}3
{ln
(1
n1)}
n=2
4
{tan1n 1
n
}5
{2n
1+2n}
6
{( nn+1
)n}
7
{1
n!
}
ExampleDetermine whether the following sequences converge or diverge.
1
{n3+n2+2n+13n3+n+5
}2 {nn2}3
{ln
(1
n1)}
n=2
4
{tan1n 1
n
}5
{2n
1+2n}
6
{( nn+1
)n}7
{1
n!
}
Outline
1 Sequences
2 Graphs of Sequences
3 Convergence of a Sequence
4 Squeeze Theorem for Sequences
Theorem (Squeeze Theorem for Sequences)
Let {an}, {bn}, {cn} be sequences such that an bn cn (for all values of nbeyond some fixed index N). If the sequences {an} and {cn} converge to L,then {bn} also converges to L.
Theorem (Squeeze Theorem for Sequences)Let {an}, {bn}, {cn} be sequences such that
an bn cn (for all values of nbeyond some fixed index N). If the sequences {an} and {cn} converge to L,then {bn} also converges to L.
Theorem (Squeeze Theorem for Sequences)Let {an}, {bn}, {cn} be sequences such that an bn cn (for all values of nbeyond some fixed index N).
If the sequences {an} and {cn} converge to L,then {bn} also converges to L.
Theorem (Squeeze Theorem for Sequences)Let {an}, {bn}, {cn} be sequences such that an bn cn (for all values of nbeyond some fixed index N). If the sequences {an} and {cn} converge to L,then
{bn} also converges to L.
Theorem (Squeeze Theorem for Sequences)Let {an}, {bn}, {cn} be sequences such that an bn cn (for all values of nbeyond some fixed index N). If the sequences {an} and {cn} converge to L,then {bn} also converges to L.
ExampleDetermine whether the following sequences converge or diverge. Assumethe starting index is 1, unless otherwise specified.
1
{cos pinn+1
}2
{sinn
n
}3
{3n+ sinn34n
}
ExampleDetermine whether the following sequences converge or diverge. Assumethe starting index is 1, unless otherwise specified.
1
{cos pinn+1
}
2
{sinn
n
}3
{3n+ sinn34n
}
ExampleDetermine whether the following sequences converge or diverge. Assumethe starting index is 1, unless otherwise specified.
1
{cos pinn+1
}2
{sinn
n
}
3
{3n+ sinn34n
}
ExampleDetermine whether the following sequences converge or diverge. Assumethe starting index is 1, unless otherwise specified.
1
{cos pinn+1
}2
{sinn
n
}3
{3n+ sinn34n
}
ExampleDetermine whether the following sequences converge or diverge. Assumethe starting index is 1, unless otherwise specified.
1
{cos pinn+1
}2
{sinn
n
}3
{3n+ sinn34n
}
In Summary
1 Sequences
2 Graphs of Sequences
3 Convergence of a Sequence
4 Squeeze Theorem for Sequences
Announcements
1 Regarding SW2 Uploaded draft of lecture notes
SequencesGraphs of SequencesConvergence of a SequenceSqueeze Theorem for Sequences
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