chapter vii (to student)

8/4/2019 Chapter VII (to Student)

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Sequences

Convergent sequences

Properties of convergent sequences

Convergent criterions

Sequences and limits

NGUYEN CANH Nam1

1Faculty of Applied MathematicsDepartment of Applied Mathematics and Informatics

Hanoi University of [email protected]

HUT - 2010

NGUYEN CANH Nam Mathematics I - Chapter 7
http://find/http://goback/


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Sequences




Agenda

1 Sequences

2 Convergent sequences

3 Properties of convergent sequences

4 Convergent criterions



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Sequences




Sequences

Definition

A sequence of real numbers is defined to be a function from the

set IN of natural numbers into the set IR. Instead of referring to

such a function as an assignment n f(n), we ordinarily use

the notation {an} or {a1, a2, a3, . . .}. Here, of course, andenotes the number f(n).

A sequence can be given different ways.

List the elements. For example, 12

,2

3,

3

4, . . .. From the

elements listed, the pattern should be clear.

Give a formula to generate the terms. For example,

an = (1)n2

n

n!. If the starting point is not specified, we use

the smallest value of n which will work.NGUYEN CANH Nam Mathematics I - Chapter 7


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Sequences




Bounded sequences and monotone sequencesBounded sequences

DefinitionA sequence {an} is said to be bounded above if there exists areal number c such that an c,n . It is bounded below if thereexists a real number d such that xn d,n. It is bounded if it isbounded above and is bounded below.



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Sequences




Bounded sequences and monotone sequencesMonotone sequences

Definition

Let {an} be a sequence.

1 {an} is said to be increasing if an an+

1for all n. If we

have an < an+1 for all n we say that the sequence is strictlyincreasing.

2 {an} is said to be decreasing if an an+1 for all n. If wehave an > an+1 for all n we say that the sequence is strictly

decreasing.3 A sequence that is either increasing or decreasing is said

to be monotone. If it is either strictly increasing or strictly

decreasing, we say it is strictly monotone.


S


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Sequences




Examples

Example

The sequence {an}, an =

1

n is decreasing, bounded belowby 0, bounded above by 1.

The sequence {an}, an = (1)n is not monotone, bounded

below by -1, bounded above by 1.

The sequence {an}, an = n2

is increasing, bounded belowby 0, unbounded above.


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Sequences




Definition

Definition

Let {an} be a sequence of real numbers and let L be a realnumber. The sequence {an} is said to converge to L, or that Lis the limit of {an}, if the following condition is satisfied.

For every positive number there exists a natural number Nsuch that if n N, then |an L| < .

In symbols, we say L = lim an or

L = limn

an.

If a sequence {an} is not converge then we said that it isdiverge.


Sequences


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Sequences




Examples

Example

1 Study the convergence of the sequence {an}, an =1n.

We think the sequence converges to 0. We want to show

that for every > 0, there exists N such that if n N then

|1n 0| < .

Indeed, if we choose N >1

then n N we have

|an 0| = |1n 0| = 1n 1N < .

2 Study the convergence of the sequence {an}, an = n2.

We see that when n go to infinity then so does an. Hence

{an} diverges.NGUYEN CANH Nam Mathematics I - Chapter 7

Sequences


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Sequences




Diverge to infinity

We say that a sequence {an} of real numbers diverges to +,we write lim

nan = +, if for every positive number M, there

exists a natural number N such that if n N, then an M.Note that we do not say that such a sequence is convergent.

Similarly, we say that a sequence {an} of real numbersdiverges to if for every real number M, there exists a

natural number N such that if n N, then an M.


Sequences


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Sequences




Results

Theorem

If a sequence converges, then its limit is unique.

Theorem

If a sequence{an} converges, then it is bounded, that is thereexists a number M > 0 such that|a

n| M for all n.


Sequences


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Sequences




Results

Theorem

Let{an} and{bn} be two sequences of real numbers with a = lim anand b= lim bn. Then

(1) The sequence{an + bn} converges, and

lim(an + bn) = lim an + lim bn = a+ b.

(2) The sequences{Can} and{C+ an}, where C is a constant, areconverge and

lim(Can) = Ca, lim(C+ an) = C+ a.

(3) The sequence{anbn} is convergent, and

lim(anbn) = lim an lim bn = ab.


Sequences


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Sequences




Results

Theorem (continue...)

(4) If all the bns as well as b are nonzero, then the sequence{1/bn}is convergent, and

lim(1

bn) =

1

lim bn=

1

b.

(5) If all the bns as well as b are nonzero, then the sequence

{an/bn} is convergent, and

lim(an

bn) =

lim an

lim bn=

a

b.


Sequences


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q




Results

Parts 1, 3 and 5 of the above theorem hold even when aand bare extended real numbers as long as the right hand side in

each part is defined. You will recall the following rules when

working with extended real numbers:

(1)+ = = ()() = (2) = () = () =

(3) If x is any real number, then(a) + x = x + = (b) + x = x =

(c)

x

=

x

= 0

(d)x

0=

if x > 0

if x < 0

(e) x = x if x > 0

if x < 0NGUYEN CANH Nam Mathematics I - Chapter 7

Sequences


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q




Result

(4) However, the following are still indeterminate forms. Their

behavior is unpredictable. Finding what they are equal torequires more advanced techniques

(a) + and (b) 0 and 0

(c)

and

0

0.


Sequences


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Results

Theorem

Let{an} be a increasing sequence of real numbers. Supposethat the sequence{an} is bounded above. Then the sequence{an} is convergent.

Analogously, if{an} is a decreasing sequence that is boundedbelow, then{an} converges.

Example

1 Consider the sequence an =1

n. We known that thissequence is decreasing and bounded below by 0. So it is

convergent.

2 (Definition of e) For n 1, define an = (1 + 1/n)n. Then

the sequence {an} is increasing and bounded above (!),whence it is convergent. (We will denote the limit of thisNGUYEN CANH Nam Mathematics I - Chapter 7

Sequences


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Squeeze Theorem

Theorem

Suppose that{an} is a sequence of real numbers and that{bn}and{cn} are two sequences of real numbers for whichbn an cn for all n. Suppose further that

limn

bn = limn

cn = L. Then the sequence{an} also converges

to L.

Example

Study the properties the sequence an = sin nn

Since 1 sin n 1 and two sequences bn =1

nand cn =

1

nconverge to 0 so the sequence {an} also converges to 0.


Sequences


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Cauchy Criterion

Definition

A sequence {an} of real numbers is a Cauchy sequence if for

every > 0, there exists a natural number N such that if n Nand m N then |an am| < .

Theorem

A sequence{an} of real numbers is convergent if and only if itis a Cauchy sequence.


Sequences

C t


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Cauchy CriterionExamples

Example

Consider {an} where an =1n

. Prove that this is a Cauchy sequence.Let > 0 be given. We want to show that there exists an integerN > 0 such that m, n> N |am an| < . That it we would like tohave

|am an| < |1

m

1

n| <

Since|

1

m

1

n| 2

. So, we see

that if N is an integer larger than2

then m, n> N |am an| < .


chapter vii (to student)

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