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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 Technologiesnamnc@mail.hut.edu.vn
HUT - 2010
NGUYEN CANH Nam Mathematics I - Chapter 7
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Sequences
Convergent sequences
Properties of convergent sequences
Convergent criterions
Agenda
1 Sequences
2 Convergent sequences
3 Properties of convergent sequences
4 Convergent criterions
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Sequences
Convergent sequences
Properties of convergent sequences
Convergent criterions
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
Convergent sequences
Properties of convergent sequences
Convergent criterions
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
Convergent sequences
Properties of convergent sequences
Convergent criterions
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.
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Sequences
Convergent sequences
Properties of convergent sequences
Convergent criterions
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
Convergent sequences
Properties of convergent sequences
Convergent criterions
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.
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Sequences
Convergent sequences
Properties of convergent sequences
Convergent criterions
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
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Sequences
Convergent sequences
Properties of convergent sequences
Convergent criterions
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.
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Sequences
Convergent sequences
Properties of convergent sequences
Convergent criterions
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.
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Sequences
Convergent sequences
Properties of convergent sequences
Convergent criterions
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.
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Sequences
Convergent sequences
Properties of convergent sequences
Convergent criterions
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.
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q
Convergent sequences
Properties of convergent sequences
Convergent criterions
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
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q
Convergent sequences
Properties of convergent sequences
Convergent criterions
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.
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Convergent sequences
Properties of convergent sequences
Convergent criterions
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
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Convergent sequences
Properties of convergent sequences
Convergent criterions
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.
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Convergent sequences
Properties of convergent sequences
Convergent criterions
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.
NGUYEN CANH Nam Mathematics I - Chapter 7
Sequences
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Convergent sequences
Properties of convergent sequences
Convergent criterions
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| < .
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