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this note is all about the subject of advanced calculus.
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CHAPTER 2CHAPTER 2
2 SEQUENCES OF REAL NUMBERS2 SEQUENCES OF REAL NUMBERS2.02.0 OBJECTIVEOBJECTIVE
At the end of this course student should be able to use the concept, definition, characteristics of a sequence to test for convergence, divergence, infimum and supremum.
2.1 INTRODUCTION2.1 INTRODUCTION
A sequence is a set of numbers written in a definite order:
The number
is called the first term,
the second term, and in general
is the nth term.
We will deal with infinite sequences and so each term an will have a successor an+1. An infinite sequence is also denoted by
that is
Example 1: Some sequences can be defined by giving a formula for the nth term.
a)
b) .
c)
In the first example, is increasing and is unbounded. In the
second example, the terms of the sequence is decreasing and
are approaching 0. In the third example, the terms of the sequence
is increasing without bound. Thus the sequences
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and are said to diverge while is said
to converge.
In general the notation
Means that the terms of the sequence approach L as n becomes large. Notice that the following definition of the limit of a sequence is very similar to the definition of a limit of a function at infinity.
Definition 1: A sequence has a limit L and we write
or
If we can take the terms as close to L as we like by taking n sufficiently
large. If exists, we say the sequence converges (or is convergent).
Otherwise, we say the sequence diverges (or is divergent).
There are sequences which are bounded but do not converge.
Example 2:
a) The terms of the sequence that is {1, -1, 1, -1, 1, -1,…}
where all the terms are bounded but the sequence does not converge.
b) The sequence is bounded but does not converge
because the terms oscillate between –1, 0 and 1.
Activity 1: Is the sequence , bounded and converges ?
2.2 BOUNDED SEQUENCE
Definition 2: The sequence is bounded if there is a positive number M
such that for all n= 1, 2, 3, ….
Example 3: The sequence is bounded because for all
positive integer n.
(i) The sequence is bounded because
2
.
(ii) The sequence
is bounded because
for all n .
(iii) Let , n 2. Is bounded ? (Activity)
(By observation, for all small positive integer. Prove that this is true for all n. Hint: Prove it by using mathematical induction).
Activity 2: Determine whether the following sequences is bounded or not.
(i) ii)
(iii) iv)
(v) vi)
General Formula for a Sequence: A sequence can be defined as a function whose domain is the set of integers
,
Example 4: Let or
.
In general, we usually write instead of the function notation for the value of the function at the number n.
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Activity 3: Find the formula for the general term of the sequences
assuming the pattern of the first few terms continues .
Remark 1: Not all sequences have a simple defining equation.
Example 5: State in general the nth term of the sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...
Solution: Observe that the first term is and . The following term is obtained by adding two consecutive terms immediately before it. It is noted as
, ,
.
Historical Notes: The sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…
is known as Fibonacci sequence, named after the Italian mathematician Leonardo of Pisa (ca. 1175-1250), also known as Fibonacci (son of Bonaccio). In addition of promoting the Hindu-Arabic numeral system, which we use today, his book Liber Abaci (1202) also contained the following problem:
“How many pairs of rabbits can be produced from a single pair in a year if every month each pair begets a new pair which from the second month on becomes productive?”
2.3 SEQUENCE DEFINED AS INFINITE ITERATION
A sequence can often be defined recursively. For example, The Fibonacci sequence is defined as:
Thus, the first few terms of the sequence above are:
1, 1, 2, 3, 5, 8, 13, 21, 34, …
The Lucas Sequence
is defined by
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Consider the sequence , where
.
It can be pictured either by plotting its terms on a number line or by plotting its graph.
It appears that the terms of the sequence are approaching 1 as n becomes large. In fact, the difference
can be made as small as we like by taking n sufficiently large. We indicate this by writing
.
In general, the notation
means that the terms of the sequence approach L as n becomes large.
0 12
1
a1 a2
a3
1 2 3 4 5 6 7
1
a7=7/8
an
n
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2.4 LIMIT OF A SEQUENCE
Definition 2: A sequence has the limit L and we write
if for every there is a corresponding integer N such that
whenever n > N .
If exists, we say the sequence converges (or is convergent),
otherwise, we say the sequence diverges (or is divergent).
In symbolic notation,
whenever n > N.
Example 6: Show that .
Solution: In this case, for given , we must show that
whenever .
Consider the difference
and solve it for n, it gives
.
Now let , then for implies
But
It implies that
whenever n>N.
As an example, if we let =0.001, then we can choose ,
and
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whenever n > 1000 .
Example 7: Show that .
Solution: Given , we must show that
whenever n>N. Observe that
If we choose , then
whenever n>N. Therefore
Example 7: Show that .
SOLUTION; Given , we must show that there exist N> 0 such that
whenever n>N. Observe that
Choose , and if n>N, then
.
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Theorem 1: If is a positive real number sequence that converges, then
is bounded.
PROOF: Let . Then for every
whenever n>N. This means,
,
for n>N or,
,
n>N. Thus
,
for n>N . Let
,
Then for all n, that is is bounded.
Note: The converse of the theorem is not true. Not all bounded sequence is convergent. For example
(i) is bounded but does not converges.
(ii) is bounded but does not converges.
Definition 3: A sequence is called increasing if
It is called decreasing if
It is called monotonic if it is either increasing or decreasing.
Example 8: The sequence is decreasing because
for all .
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Activity 4: Show that the sequence is decreasing.
Example 9: Define a sequence as:
.
The sequence can be shown to be bounded and monotonic. Let .
Then
or
or
which has the solutions . Given sequence is not negative therefore meaning
.
Activity 5: A sequence is defined as:
.
Show that the sequence is monotonic (increasing) and for all n. Then, show that the limit is 2.
Theorem 2: Every bounded, monotonic sequence is convergent.
Proof. Exercise
Note: We write when for every M > 0, N > 0 for
.
Also, when for every M > 0,
N > 0 .
In both cases, is said to be convergent.
Theorem 3: Every unbounded sequence is divergent.
Definition 4: A sequence is called a Cauchy sequence if for every >0 there exist N > 0 such that
whenever m, n > N .
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Theorem 3: A sequence is convergent if and only if it is Cauchy.
PROOF Suppose , and given > 0
when n > N. Thus
when m, n > N .
Theorem 4: The limit of a sequence, if it exists, is unique.
PROOF: Suppose exist and assume there are two limits, and where .
Choose . Therefore, there exist such that
whenever
and
.
Choose N = max thus
whenever which is false. Thus, .
Example 10: Is the sequence converges? Is the sequence
monotonic?
SOLUTION: By looking at the ratio of the terms and ,
Therefore, the sequence is decreasing. Is the sequence bounded? (Activity)
10
Example 11: Determine whether exist.
SOLUTION: If we define and , therefore is in
indefinite form, that is when n tends to infinity. Therefore the L’Hopital
rule can be applied, thus
Example 10 Find if it exist.
SOLUTION: The limit of the sequence can be determine by applying the L’Hopital rule repeatedly.
.
Activity 6: Show that .
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2.5 Limit Laws for Sequences
Theorem
Let and . Therefore
(i) .
(ii)
(iii)
PROOF
(i)
whenever .
for the same , whenever
Let N = max . Thus
whenever n > N .
Activity
Prove part (ii) and (iii).
2.6 An Important Note on e
Observe that
Replacing ,
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Therefore
Notice that . Furthermore,
.
Thus, .
(i) Let x be a positive number and let be the greatest integer that is less than or equal to x, that is n = [x]. Therefore
n x n + 1,
and
.
But
( see the example below)
Therefore for all positive integer x.
Example
If , show that
PROOF Let . Then
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Also,
2.7 The Concept of Supremum and
Infimum
Suppose A is a set of real numbers. A number M is the upper bound of A if for all x in A. If is another upper bound of A, then is called the
least upper bound of A if M for all M of A. is called the supremum of A or the least upper bound and is written:
.
In a similar manner m is the lower bound of A if for all x in A; is the greatest lower bound or the infimum of A if for all m of A, and is written
.
The Completeness Axiom
Let A be a nonempty set of real numbers. If A has an upper bound, then A has a least upper bound, called the supremum of A and denoted by sup A. If A has a lower bound, then A has a greatest lower bound, called the infimum of A and denoted by inf A.
Example
(i) A = Interval[0,1] ; ,
(ii) A = Interval (0,1]; ,
(iii)
(iv)
(v) If A is the single point, then
.
Note: In example (i) the sup A and inf A are in A, but in (ii) inf A = 0 is not in the interval (0, 1]. In the set A = interval (0, 1) both the sup A and inf A are outside the interval.
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2.8 Limit superior and Limit inferior
Suppose is as sequence of real numbers which is bounded.
Defined
.
and
Then the sequence is bounded and decreasing and is bounded
and increasing. If the limit of , , exist, then it is defined as limit
superior of the sequence ,
.
If the limit of exist, then it is defined as limit inferior of the sequence
Example
Determine the and of the following sequence:
(i) .
Suppose
and
then
Thus,
.
In a similar manner,
thus and
15
.
(ii)
Thus
Therefore, .
thus
and
(iii)
…
therefore
.
n = 1, 2, 3, …
.
16
In a similar manner,
,
Therefore,
,
n = 1, 2, 3, …
Therefore .
Notation
Suppose is an unbounded sequence of real numbers.
If , then we defined .
If , the we defined .
A bounded sequence doesn’t necessarily have a limit. Nevertheless, the sequence always has and .
(i) The sequence
= {1, -1, 1, -1, 1, -1, …}
is bounded but is divergent. Nevertheless, the and the .
Definition
whenever n > N .
whenever n > N .
Theorem Let be a sequence of real numbers and . Then
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Example
.
PROOF
Suppose
whenever
This means,
whenever or
whenever .
Therefore
whenever or
whenever , which means
whenever . Therefore
.
Activity
The case for can be shown in a similar manner.
; n =1,2,3, …
The sequence is alternating between the value of +, 0, - for . When n
tends to infinity, the sequence tends to 0. Therefore, . From the theorem above,
.
Theorem Let be a bounded sequence and
.
Therefore, .
PROOF
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Example
whenever , i.e.
whenever
or .
Therefore whenever .
Also,
for the same ,
i.e.
or
Therefore
.
Choose N = max
whenever
or whenever . Thus
.
2.9 Sequence of Functions
Misalkan , n = 1, 2, 3, … adalah jujukan fungsi nyata dengan x berada dalam selang I. Jujukan fungsi ini dikatakan menumpu ke suatu fungsi f untuk setiap x dalam I jika untuk setiap > 0 wujud N > 0 sedemikian hingga apabila n > N.
Umumnya, nombor integer N yang dimaksudkan di sini bergantung kepada x dan . Kita tulis N = N(x, ) . Penumpuan ini disebut sebagai penumpuan titik demi titik, dan diringkaskan : (titik demi titik) . Akan tetapi, apabila N yang diperoleh tidak bergantung kepada nilai x, tetapi hanya
19
bergantung kepada , maka penumpuan tersebut dinamakan penumpuan seragam, dan diringkaskan : (seragam) .
Perhatikan contoh yang berikut untuk membezakan antara kedua-dua jenis penumpuan tersebut:
Contoh Tunjukkan bahawa menumpu seragam ke 0 dalam
selang terbuka .
Penyelesaian
Perhatikan bahawa
,
dan
jika dan hanya , atau
atau
(sebab )
Tetapi ,
Andainya kita diberi > 0 , pilih
maka apabila n > N() untuk semua x dalam
selang .
Untuk semua x dalam selang ini, N hanya bergantung kepada , tidak bergantung kepada x, maka penumpuan adalah seragam.
Contoh 2Tunjukkan bahawa , n = 1, 2, 3, … tidak menumpu seragam kepada 0 dalam selang terbuka (0, 1).
Penyelesaian
20
untuk x(0,1). Apabila
maka .
Justeru itu, tidak mungkin wujud N > 0 sedemikian hingga
Maka, tidak menumpu seragam dalam (0, 1) .
The Completeness Axiom
(Axiom Kelengkapan)
Let S be a nonempty set of real numbers. If S has an upper bound, then S has a least upper bound, called the supremum of S and denoted by sup S. If S has a lower bound, then S has a greatest lower bound, called the infimum of S and denoted by inf S.
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