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)

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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

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.

(ii)

Thus

Therefore, .

thus

and

(iii)

…

therefore

.

n = 1, 2, 3, …

.

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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

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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

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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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