Sequences of Real Numbers Part III

Advanced Calculus Unit XIX

sequences of real numbers – Part III

Objectives

From this session a learner is expected to achieve the following

Introduce the concept of monotone sequences
Study that a bounded monotonic sequence is convergent
Learn that monotone convergence theorem helps in finding a sequence of real numbers that converges to the positive square root of a positive real number.
Construct a sequence that converges to the transcendental number e.

Contents

1. Introduction

2. Montone Sequences

3. The Calculation of Square Roots

4. Euler Number

1. Introduction

In the previous session (Sequences of Real Numbers -Part II) we have seen that every convergent sequence is bounded, but the converse is not true. In this session, we search for a condition which ensures the convergence of a bounded sequence; and end up with the Monotone Convergence Theorem that establishes the existence of the limit of a bounded monotone sequence. We begin with the definition of monotone sequence and discuss associated results and examples. We will see that Monotone Convergence Theorem is helpful in finding limit of certain seqeunces. It helps us in finding a sequence of real numbers that converges to the positive square root of a positive real number. A discussion on Euler Number will be made in this session.

2. Montone Sequences

Definition Let be a sequence of real numbers. We say that the sequence is increasing(or monotonically increasing) if it satisfies the inequalities

We say that is decreasing(or monotonically decreasing)if it satisfies the inequalities

We say that is monotone if it is either increasing, or it is decreasing.

The following sequences are increasing:

if .

The following sequences are decreasing:

The following sequences are not monotone:

The following sequences are not monotone, but they are ultimately monotone:

Example 1 Show that the sequence where

is monotonically decreasing.

Solution

We note that for

Hence

Therefore,

for

and hence the given sequence is monotonically decreasing.

Theorem 1 (Monotone Convergence Theorem) A monotone sequence of real numbers is convergent if and only if it is bounded. Also,

(a) If is a bounded increasing sequence, then

(b) If is a bounded decreasing sequence, then

Proof. First of all we note that a convergent sequence is bounded. Hence, in particular, a bounded monotone sequence of real numbers is also convergent.

Conversely, let be a bounded monotone sequence. Then is either increasing or decreasing.

(a)We first treat the case where is a bounded increasing sequence. By hypothesis, there exists a real number such that for all Thus the subset of the set of real numbers is bounded above. Hence by the Supremum Property of real numbers, supremum of the set exists. Let the supremum be

Claim

If is given, then is not an upper bound for the set , and hence there exists a natural number such that is a member of the set and . But since is an increasing sequence it follows that

for all

Therefore it follows that for all

i.e., for all

Since is arbitrary, we have converges to .

(b) If is a bounded decreasing sequence, then it is clear that is a bounded increasing sequence. We have seen in part (a) that . On the other hand, ; and also, we have

Therefore

This completes the proof.

RemarkThe above theorem gives us a way of calculating the limit of the sequence provided we can evaluate the supremum in case (a), or the infimum in case (b).

Example 2.

Solution

Obviously the sequence is decreasing and bounded (1 is a bound for the sequence). Hence by part (b) of Theorem 1,

…(1)

Clearly 0 is a lower bound for the set , and also 0 is the infimum of the set ; hence (1) implies .

Example 3Examine the convergence of the sequence , where

for .

Solution

It can be seen that

and hence is monotonically increasing.

Obviously the sequence is bounded below by

Also,

Hence the sequence is bounded above by 1.

We conclude that the sequence is a bounded monotone sequence and hence is convergent.

Example 4Examine the convergence of the sequence , where

for .

Solution

Since , we see that is an increasing sequence. Hence by the Monotone convergence Theorem the question of whether the sequence is convergent or not is reduced to the question of whether the sequence is bounded or not.

We note that

Hence the sequence is unbounded, and therefore by Monotone convergence Theorem is divergent.

Example 5 Let be defined inductively by for . Show that .

Solution

Direct calculation shows that . Hence we have . We show, by induction, that for all . We have already noted that this is true for . If holds for some , then

so that . Therefore for all .

We now show, by induction, that for all . The truth of this assertion has been verified for . Now suppose that for some ; then

and hence it follows that

Thus implies that . Therefore for all .

We have shown that the sequence is increasing and bounded above by 2. It follows from the Monotone convergence Theorem that the sequence converges to a limit that is at most 2. In this case it is not so easy to evaluate by calculating . Hence we proceed as follows:

Let Then limit of the 1-tail of also has the limit .

Since for all , we have the limit of the sequence and the limit of the sequence are the same. Hence, we obtain

Hence it follows that .

Example6 Let be such that Let be the sequence of real numbers defined by

Show that

Solution

We first prove that is bounded above by i.e., we have to show that We prove this by Principle of Mathemtical Induction.

The result is true for becuase

As induction argument, assume the result is true for i.e., suppose

Then

since

Hence

and this shows that Hence is bounded above by

Also,

since

Hence

so that

Hence is monotonic increasing.

Being monotonic increasing and bounded, the sequence is convergent. Let Then, also,

Since

we have

implies

i.e.,

Example7 Let be the sequence of real numbers defined by for . Show that

Solution

Note that and ; hence . We claim that the sequence is increasing and bounded above by 2. To show this we will show, by induction, that for all . This fact has been verified for . Suppose that it is true for ; then , and hence it follows that

Noting that , the above implies

Hence the validity of the inequality , implies the validity of . Therefore for all .

Since is a bounded increasing sequence, it follows from the Monotone convergence Theorem that it converges. Let the limit of be .

The relation gives which implies .

Hence which has the roots . Since the terms of all satisfy , it follows that we must have Therefore .

M 2

3. The Calculation of Square Roots

Now we show that Monotone Convergence Theorem helps us in finding a sequence of real numbers that converges to the positive square root of a positive real number.

Let ; we will construct a sequence of real numbers that converges to .

Let be arbitrary and define for . We now show that the sequence converges to .

We first show that for . Since satisfies the quadratic equation , this equation has a real root. Hence the discriminant must be nonnegative; that is for .

To see that is ultimately decreasing, we note that for we have

Hence, for all . It follows from the Monotone Convergence Theorem that exists. Moreover, it follows that the limit must satisfy the relation

whence it follows that

or or .

Thus .

For the purposes of calculation, it is often important to have an estimate of how rapidly the sequence converges to . As above, we have for all , whence it follows that . Thus we have

for .

Using this inequality we can calculate to any desired degree of accuracy.

Mod 3

4. Euler’s Number

We now construct a sequence that converges to one of the most important transcendental numbers in mathematics.

For this, let for . We will now show that the sequence is bounded and increasing; hence it is convergent.

If we apply the Binomial Theorem, we have

If we divide the powers of into the terms in the numerators of the binomial coefficients, we get

Similarly, we have

Note that the expression for contains terms, while that for contains terms. Moreover, each term appearing in is less than or equal to the corresponding term in , and has one more positive term. Therefore we have , so that the terms of are increasing.

To show that the terms of the sequence are bounded above, we note that if then . Moreover so that . Therefore, if , then we have

Being the finite sum ofgeometric progression with common ratio we have

and hence we deduce that for all . The Monotone Convergence Theorem implies that the sequence converges to a real number. Also, since for all , the limit of lies between 2 and 3. We define the number to be the limit of the sequence. The limit of the sequence is the famous Euler number e, whose approximate value is which is taken as the base of the natural logarithm.

Summary

In this session, we have described Monotone Convergence Theorem and used it for finding limits of certain sequences. We have seen that it is helpful in finding a sequence of real numbers that converges to the positive square root of a positive real number. A discussion on Euler Number have been made in this session.

Assignments

1. Let and let . Define for . Show that converges and find the limit.

2. Let be a bounded sequence, and for each let and . Prove that and are convergent. Also prove that if , then is convergent.

3. Let be an infinite subset of that is bounded above and let . Show there exists an increasing sequence with for all such that .

4. Let for each . Prove that is increasing and bounded, and hence converges.

5. Let and for . Show that is bounded and monotone. Find the limit.

6. Let where and and for . Find

QUIZ

1.Let and for . Then pick the true statement.

(a) is monotone, but not bounded.

(b) is bounded, but not monotone. .

(d) is convergent.

Ans. (d) is convergent.

2.Let and . Then pick the true statement.

(a) is monotonically decreasing

(b) is monotonically increasing

(d) none of the above.

Ans. (b) is monotonically increasing

3.Limit of the sequence is ______

(a)

(b)

(c)

(d)

Ans. (a)

4.Limit of the sequence is ______

(a)

(b)

(c)

(d)

Ans. (b)

FAQ

1. State Supremum Property of real numbers.

Answer:

The Supremum Property (or Completeness Property) of is the following: Every nonempty set of real numbers that has an upper bound has a supremum in .

2. Define transcendental number.

Answer:

A transcendental number is a real or complex number, which is not a root of a non-constant polynomial equation with rational coefficients. and e are transcendental numbers.

3. A careless assumption that a sequence is ‘convergent’ leads to aburd conclusions. Comment.

Answer:

The issue of convergence must not be ignored or casually assumed. The following example illustrates this:

Consider the sequence defined by Assuming the ‘convergence’ (actually wrong! The sequence is not convergent) with we would obtain so that Of course, this is absurd as always and hence must be This absurdity is due to the wrong assumption that the sequence is convergent. Hence, it is required to examine the convergence of the sequence before finding its limit.

Glossary

Increasing Sequence: A sequence of real numbers is increasing (or monotonically increasing) if it satisfies the inequalities

Decreasing Sequence: A sequence of real numbers is decreasing (or monotonically decreasing) if it satisfies the inequalities

Monotone Sequence: A sequence of real numbers is monotone if it is either increasing, or it is decreasing.

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.