M38 Lec 111413

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

Outline

1 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, ...}


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, ...}


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?


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}


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}


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}


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}


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}


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}


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}


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


{3n8}n=1

2{12 , 14 ,18 , 116 , ...}=

{(1)n2n

}n=1

3 {1,0,1,0, ...}

{an}n=1 where an =



{3n8}n=1

2{12 , 14 ,18 , 116 , ...}= {

(1)n2n

}n=1

3 {1,0,1,0, ...}

{an}n=1 where an =


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}


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}


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}


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}


13 ,

14 , ...}= {

1

n

}n=1

2 {1,12 , 13 ,14 , ...}= {(1)n+1

n

}n=1

3 {12 ,23 ,

34 ,

45 , ...}=

{ nn+1

}


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




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{(1)n2n

}n=1

0

Example{1,0,1,0, ...}

0

Example{ 1n

}n=1

0

Example{(1)n+1

n

}n=1

0

Example{ nn+1}n=1

0

Outline

1 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




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




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


5

{(1)n+1

n


6{ nn+1}n=1 converges to 1.

Example

1 {3n8}n=1 is divergent.

2

{(1)n2n


3 {1,0,1,0, ...} is divergent.4{ 1n


5

{(1)n+1

n



Example

1 {3n8}n=1 is divergent.2

{(1)n2n

}n=1

converges to 0.

3 {1,0,1,0, ...} is divergent.4{ 1n


5

{(1)n+1

n



Example


{(1)n2n


3 {1,0,1,0, ...} is divergent.4{ 1n


5

{(1)n+1

n



Example


{(1)n2n


3 {1,0,1,0, ...}

is divergent.4{ 1n


5

{(1)n+1

n



Example


{(1)n2n


3 {1,0,1,0, ...} is divergent.

4{ 1n


5

{(1)n+1

n



Example


{(1)n2n


3 {1,0,1,0, ...} is divergent.4{ 1n

}n=1

converges to 0.

5

{(1)n+1

n



Example


{(1)n2n


3 {1,0,1,0, ...} is divergent.4{ 1n


5

{(1)n+1

n



Example


{(1)n2n


3 {1,0,1,0, ...} is divergent.4{ 1n


5

{(1)n+1

n

}n=1

converges to 0.


Example


{(1)n2n


3 {1,0,1,0, ...} is divergent.4{ 1n


5

{(1)n+1

n



Example


{(1)n2n


3 {1,0,1,0, ...} is divergent.4{ 1n


5

{(1)n+1

n


6{ nn+1}n=1

converges to 1.

Example


{(1)n2n


3 {1,0,1,0, ...} is divergent.4{ 1n


5

{(1)n+1

n



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


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!)


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!)


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

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!

}


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!

}


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!

}


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!

}


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!

}


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!

}


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!

}


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




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

}


1

{cos pinn+1

}

2

{sinn

n

}3

{3n+ sinn34n

}


1

{cos pinn+1

}2

{sinn

n

}

3

{3n+ sinn34n

}


1

{cos pinn+1

}2

{sinn

n

}3

{3n+ sinn34n

}

In Summary

1 Sequences




Announcements

1 Regarding SW2 Uploaded draft of lecture notes

SequencesGraphs of SequencesConvergence of a SequenceSqueeze Theorem for Sequences

M38 Lec 111413

Documents

Transcript of M38 Lec 111413