Advanced Calculus Unit Xvii

8.1.2013

ADVANCED CALCULUS UNIT XVII

sequences of real numbers – Part I

Objectives

From this session a learner is expected to achieve the following

Introduce the concept of sequence of real numbers
Study the limit of a sequence and familiarise with examples
Study that a sequence converges if and only any tail of the sequence also converges.

Contents

1. Introduction

2. Sequences of Real Numbers

3. The Limit of a Sequence

4. Tails of Sequences

Introduction

In this session we familiarize the concept of sequence of real numbers and convergence of sequences. From the definition of limit of a sequence we will observe that the convergence (or divergence) of a sequence depends only on the ultimate behavior of the terms of the sequence. Hence for any natural number , if we drop the first terms of the sequence, then the resulting sequence converges if and only if the original sequence converges, and in this case, the limits are the same. We will state this formally as a theorem after introducing the concept of a tail of a sequence.

Sequences of Real Numbers

Definition Afunction from the set of natural numbers to a nonempty set S is called a sequence in the set S.

In this session we are interested in a particular, but important, case when. Definition follows:

Definition A sequence of real numbers (or a sequence in) is a function on the set of natural numbers whose range is contained in the set of real numbers.

Remark Being a function from the set of natural numbers to the set of real numbers, a sequence in assigns to each natural number a uniquely determined real number. The real numbers so obtained are called the elements of the sequence, or the values of the sequence, or the termsof the sequence. It is customary to denote the element of assigned to by a symbol such as (or , or ). Thus, if is a sequence, we shall ordinarily denote the value of at by , rather than by . We will denote this sequence by the notations

We use the parentheses to indicate that the ordering induced by the natural order of is a matter of importance. Thus, we distinguish notationally between the sequence, whose terms have an ordering, and the set of values of this sequence which are not ordered. For example, the sequence is and alternates between and 1, whereas is equal to the set .

In defining sequences it is often convenient to list the terms of a sequence in order, stopping when the rule of formation seems evident. Thus we may write

for the sequence of even natural numbers, or

for the sequence of reciprocals of the natural numbers, or

for the sequence of reciprocals of the squares of the natural numbers. A more satisfactory method is to specify a formula for the general term of the sequence, such as

Some times it is convenient to specify the value and a formula for obtaining when is known. Still more generally we may specify and a formula for obtaining from . We refer to either of these methods as an inductive or recursive definition of the sequence. In this way, the sequence of even natural numbers can be defined by

;

or by the definition

Remark Sequences that are given by an inductive process often arise in computer science. In particular, sequences defined by an inductive process of the form given, for are especially amenable to study using computers. Sequences defined by the process: given, for , can also be treated.

Example 1 If , the sequence, all of whose terms equal , is called the constant sequence. Thus the constant sequence 1 is the sequence , all of whose terms equal 1.

Example 2 The sequence of squares of the natural numbers is the sequence , which, of course, is the same as the sequence .

Example 3 If , then the sequence is the sequence . In particular, if , then we obtain the sequence

.Example 4 The Fibonacci sequence is given by the inductive definition

The first six terms of the Fibonacci sequence are seen to be .

We now introduce some important ways of constructing new sequences from given ones.

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The Limit of a Sequence

There are a number of different limit concepts in real analysis. The notion of limit of a sequence is the most basic.

Definition Let be a sequence of real numbers. A real number is said to be a limit of if for every there exists a natural number such that for all , the terms satisfy

If is a limit of the sequence, we also say that converges to (or has a limit). If a sequence has a limit, we say that the sequence is convergent; if it has no limit, we say that the sequence is divergent.

When a sequence has a limit in , we will use the notation

Also we will sometimes use the symbolism , which indicates the intuitive idea that the values approach the number as .

RemarkThe notation is used to note that the choice of depends on the value of ; however, it is often convenient to write instead of . In many cases, a small value of will usually require a large value of in order to guarantee that the distance between and is less than for all .

Example 5 Show that.

Solution

Here and . We have to show that for every there is a natural

number such that if , then

.…(1)

To show this, we note thatby the Archimedean Property, corresponding to any there is a natural number exceeding . Now if is a natural number with , then for any such that , we will have so that . Thus (1) is proved, and hence

Example 6.

Solution

Here and . We have to show that for every there is a natural number such that if , then

.…(2)

By the Archimedean Property corresponding to any there is a natural number such that . Then if then so that and hence it follows that . This proves (2) and hence .

Example7 The sequence does not converge to 0. Can it converge to any other real number?

Solution

Here .

If we choose , then, for any natural number, one can always select an even number , for which the corresponding value and for which . Thus, the number 0 is not the limit of the given sequence.

The sequence doesn’t converge to 2, because if we choose , then, for any value of , one can always select an odd number , for which the corresponding value and for which . Thus, the number 2 is not the limit of the given sequence.

Also, because the terms of the sequence are alternately 0 and 2, they never accumulate near any other value. Therefore, the

sequence diverges.

Example 8 Show that the sequence is divergent.

Solution

Take a positive smaller than 1 so that the bands shown in Fig.1 about the lines and do not overlap. Any will do. Convergence to 1 would require every point of the graph beyond a certain index K to lie inside the upper band, but this will never happen. As soon as a point lies in the upper band, the next point will lie in the lower band. Hence the sequence cannot converge to 1. Likewise, it cannot converge to . On the other hand, because the terms of the sequence get alternately closer to 1 and , they never accumulate near any other value. Therefore, the sequence diverges.

Theorem 1 (Uniqueness of Limits) A sequence of real numbers can have at most one limit.

Proof.

Suppose that and are both limits of For each there exist such that

for all ,

and there exists such that

for all .

We let be the larger of and . We recall that, by triangle inequality, for any real numbers a and b,

Hence, in our present case, for we have

Now we recall the result: “If is such that for every positive , then .”

Hence, in our present case, since is an arbitrary positive number, and since we conclude that implies or

This completes the proof.

Remark In the light of the above theorem, hereinafter we will say ‘the limit’ of a sequence instead of ‘a limit’ of a sequence.

DefinitionFor and , the neighborhood of is the set

It can be verified that is equivalent to . Using this, the definition of convergence of a sequence can be formulated in terms of neighborhoods as follows:

Definition Let be a sequence of real numbers. A real number is said to be a limit of if for every there exists a natural number such that for all , the terms belong to the -neighborhood .

The following theorem gives various equivalent statements saying that a sequence converges to .

Theorem2 Let be a sequence of real numbers, and let . The following statements are equivalent.

(a) converges to .

(b) For every , there is a natural number such that for all , the terms satisfy .

(d) For every -neighborhood of x, there is a natural number such that for all , the terms belong to .

Proof. By the definition, the equivalence of (a) and (b) follows. The equivalence of (b), (c), and (d) follows from the following implications:

This completes the proof.

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4. Tails of Sequences

Definition If is a sequence of real numbers and if is a given natural number, then the -tail of is the sequence

For example, the 2-tail of the the sequence , is the sequence .

Theorem3 Let be a sequence of real numbers and let . Then the m-tail of converges if and only if converges. Also, in this case,

Proof. First we observe that for any , the th term of is the the term of . Similarly, if , then the th term of is the th term of .

Assume that sequence converges to . Then given any , if the terms of for satisfy , then the terms of for satisfy Thus we can take so that also converges to .

Conversely, if the terms of for satisfy , then the terms of for satisfy . Thus we can take .

Therefore, the sequence converges to if and only if converges to . This completes the proof.

Definition A sequence ultimately has a certain property, if some tail of has this property.

For example, the sequence is ultimately constant. On the other hand, the sequence is not ultimately constant.

Remark (The notion of convergence in terms of ultimate property) A sequence converges to if and only if the terms of are ultimately in every -neighborhood of .

Theorem4 Let be a sequence of real numbers and let . If is a sequence of positive real numbers with and if for some constant and some we have

for all

then it follows that

Proof

It is given that Hence corresponding toa given , there exists such that implies

Hence it follows that if both and then

Since is arbitrary, we obtain that . This completes the proof.

Example 9 If , then show that .

SolutionSince , we can write , where so that . By Bernoulli’s Inequality we have . Hence

Since Theorem 4 with and implies .

Example 10 If , then show that .

Solution

Case 1) When, is the constant sequence and it evidently converges to 1.

Case 2) If , then for some . Hence by Bernoulli’s Inequality

for.

Therefore we have

so that

Consequently we have

for.

Since, Theorem 4 with and implies when .

Case 3) If c is such that ; then for some . Hence by Bernoulli’s Inequality it follows that

from which it follows that for . Therefore we have

so that

for.

Since Theorem 4 with and implies when .

Example11Show that .

Solution

We note that for . Hence we can write for some when . Hence for . By the Binomial Theorem, if we have

and hence it follows that

Hence for . Now if is given, it follows from the Archimedean Property that there exists a natural number such that . It follows that if then , and hence it follows that

Since is arbitrary, we conclude that

Summary

In this session the concept of sequence of real numbers and convergence of sequences have been introduced. From the definition of limit we have observe that the convergence (or divergence) of a sequence depends only the ultimate behavior of the terms of the sequence. We have stated this formally as a theorem after introducing the concept of a tail of a sequence.

Assignments

1. For any , prove that .

2. Using the definition of the limit of a sequence, establish that .

3. Show that

4. By means of an example, show that the convergence of need not imply the convergence of .

5. Prove that if and only if .

6. Show that .

Quiz

1. Which one of the following is not a sequence?

(a) (1, 1, 1, 1,1, … )

(b) (1, 3, 5, 7, . . . , (), . . . )

(c)

(d)

Ans. (d)

2. The first five terms of the sequence where , are ______

(a) 1, 1, 1, 1,1

(b)

(c)

(d) none of these

Ans. (d) none of these

3.nth term of the sequence is ______

(a)

(b)

(c)

(d)

Ans. (b)

4. The limit of the sequence is

(a) 1

(b) 2

(d) 4

Ans. (a) 1

5. The sequence ______

(a) converges to 0

(b) converges to 1

(d) diverges

Ans. (d) diverges

6. If , then ______

(a) 0

(b) 1

(d) doesn’t exist

Ans. (a) 0

FAQ

1. What is the statement of Bernoulli’s Inequality?

Answer. If , then

for all .

2. What is the statement of the Binomial Theorem?

Answer.

The expansion of when n is a positive integer is given by

3. Can we predict the limit of a convergent sequence by the definition of limit discussed in this session?

Answer The definition of the limit of a sequence of real numbers is used to verify that a proposed value is indeed the limit. It does not provide a means for initially determining what that value of x should be. But quite often it is necessary in practice to arrive at a conjectured value of the limit by direct calculation of a number of terms of the sequence. Computers can be very helpful in this respect, but since they can calculate only a finite number of terms of a sequence, such computations do not in any way constitute a proof, but only a conjecture, of the value of the limit.

Glossary

Sequence: A function from the set of natural numbers to a nonempty set S is called a sequence in the set S.

In this session we are interested in a particular, but important, case when . Definition follows:

Sequence of real numbers: A sequence of real numbers (or a sequence in ) is a function on the set of natural numbers whose range is contained in the set of real numbers.

Limit of a Sequence: Let be a sequence of real numbers. A real number is said to be a limit of if for every there exists a natural number such that for all , the terms satisfy

Convergent Sequence: If a sequence has a limit, the sequence is convergent.

Divergent Sequence: If a sequence has no limit, the sequence is divergent.

References:

1. T. M. Apostol, Mathematical Analysis, Narosa Publishing House, New Delhi, 1985.

2. R. R. Goldberg, Real Analysis, Oxford & I.B.H. Publishing Co., New Delhi, 1970.

3. D. Soma Sundaram and B. Choudhary, A First Course in Mathematical Analysis, Narosa Publishing House, New Delhi, 1997.

4. Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, Wiley India Pvt. Ltd., New Delhi.