Advanced Calculus Unit XXV

Advanced Calculus Unit XXV

Alternating series - Part i

Objectives

From this session a learner is expected to achieve the following

· Familiarise with alternating series and divergence or convergence of it.

· Study Leibniz test and familiarise it through examples

· Learn to approximating sums of alternating series.

1. Introduction

In this session we discuss series with alternatively positive and negative terms and discuss their convergence. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges. We know that a series with all its terms nonnegative either converge or diverge but never oscillate. But an alternating series either converges or oscillate. Leibniz test for verifying the convergence of alternating series will be discussed. Theorem on approximating sums of alternating series also will be discussed.

2. Alternating series

Definition (Alternating series) A series in which the terms are alternatively positive and negative is an alternating series.

Example 1 The following are examples of alternating series.

;

The alternating series

is called the alternating harmonic series.

Notation An alternating series may be written as where each is positive and the first term is positive. If the first term in the series is negative, then we write the alternating series as .

Example 2 Test the convergence of the alternating series

Solution

To examine the convergence of the given alternating series, it is enough to consider the convergence of the associated sequence of partial sums. For this, let be the sum of the first n terms of the series. Then

Here the sequence of nth partial sums is 3, 1, 0, 3, 1, 0 . . . which doesn’t converge; also it doesn’t tend to as . Therefore, the given alternating series is oscillatory.

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

3. Leibniz test

The following theorem known as Leibniz Test or the alternating series test tells us that an alternating series will converge if the terms converge to 0 monotonically.

Theorem 1 (Leibniz Test or Alternating Series Test)

Suppose is a sequence of positive numbers such that

(a) (i.e., the ’ s decrease) and

(b) ,

Then the alternating series is convergent.

In the proof the theorem we need the following result.

Theroem 2 A sequence converges to a limit L if and only if the sequences of even-numbered terms and odd-numbered terms both converge to L.

Using Theorem 2, we now prove Theorem 1.

Proof of Theorem 1

If n is an even integer, say then the sum of the first n terms is

The first equality shows that is the sum of m nonnegative terms, since, by assumption (a), each term in parentheses is positive or zero. Hence , and the sequence of even numbered partial sums is nondecreasing. The second equality shows that . Since is nondecreasing and bounded from above, by non decreasing Sequence Theorem, it has a limit, say

. . . (1)

If n is an odd integer, say then the sum of the first n terms is . Since, by assumption (b), , in particular we have

and, hence as ,

. . . (2)

Combining the results of (1) and (2) , we have the sequence of even-numbered partial sums and the sequence of odd-numbered partial sums both converge to L. Hence by Theorem 2, the sequence of partial sums also converge to L. i.e.,

As the sequence of nth partial sums of the given series converges, the given series converges. This completes the proof.

Remark Leibniz Test works for series of the form also. i.e., if is a sequence of positive numbers such that

(a) (i.e., the ‘ s decrease) and

(b) ,

Then the alternating series is convergent.

Remark A common mistake is to try to apply the conditions of the Leibniz Test, and then, upon discovering that some condition doesn't hold, conclude that the series diverges. The theorem only says that if the conditions are true, then the series converges; it does not say what happens if the conditions are not true. On the other hand, if , we can conclude that the series diverges, by the Zero Limit Test of divergence of series.

Example 3 Show that the alternating Harmonic series

is convergent.

Solution

The given series is an alternating series of the form where .

Since , we have .

Also

Hence all the conditions of Leibniz Test are satisfied by the given alternating series and so it is convergent.

Example 4 Test the convergence of the series

Solution

The given series is an alternating series of the form where

Thus and .

Also

Thus all the conditions of Leibniz’s test are satisfied by the given alternating series and so it is convergent.

Example 5 Test the convergence of

where x and a are positive real numbers.

Solution

The given series is an alternating series of the form where .

Since a > 0, we have

. . . (3)

As all the terms in (3) are positive, we have

Hence the terms of the given alternating series are in the decreasing order.

Also, as x and a are positive real numbers, we have

Hence for the given alternating series all the conditions of Leibniz’s Test are satisfied and so it is convergent.

Example 6 Test the convergence of

Solution

The given series is an alternating series of the form where

, , , . . .

with

In general (for ) with

Also

Hence all the conditions of Leibniz’s Test are satisfied by the given alternating series and so it is convergent.

Example 7 Discuss the convergence of the series

Solution

The given series is an alternating series of the form , where

Hence and , since .

Hence the terms are in the decreasing order.

Also

Hence all the conditions of Leibniz’s Test are satisfied by the given alternating series and so it is convergent.

Example 8 Test the convergence of

Solution

The given series is an alternating series of the form , where

Now

Hence or

and therefore the terms of the given alternating series are in the decreasing order.

Also

Hence by the Leibniz’s Test, the given series is convergent.

Example 9 Use Leibniz’s Test for the convergence of an alternating series to examine the convergence of the series

Solution

The given series is an alternating series of the form , where

Hence, and and thus by Leibniz’s test, the given series is convergent.

Example 10 Examine the convergence of the series:

Solution

Writing the series in the form , we have and

, since .

Thus

Also, since , .

Hence all the conditions of Leibniz test are satisfied and hence the given series converges.

Example 11 Examine the convergence of the series:

Solution

The given series is an alternating series of the form where

and

Hence

Since x is positive and less than 1,

Hence for all n.

Since with as , we have

Hence all the conditions of Leibniz test are satisfied and the series is convergent.

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXxxx

4. Approximating Sums of Alternating Series

Theorem 3: (The Alternating Series Estimation Theorem)

If an alternating series of either of the form

satisfies the hypotheses of the Leibniz test, and if L is the sum of the series, then:

(a) L lies between any two successive partial sums; i.e., either

depending on which partial sum is larger.

(b) If L is approximated by , then the absolute error satisfies

Proof. We prove the theorem only for series of the form as the case of the series of the form can be dealt similarly.

Suppose the series satisfies the hypotheses of the Leibniz test, and if L is the sum of the series, then by Theorem 2 both the odd-numered partial sums and the even-numbered partial sums converge to L. Also, since

and

we have, the odd-numbered partial sums form a decreasing sequence converging to L and the even-numbered partial sums form an increasing sequence converging to L. Thus the successive partial sums oscillate from one side of L to the other in smaller and smaller steps with the odd-numbered partial sums being larger than L and the even-numbered partial sums being smaller than L. Thus depending on whether n is even or odd, we have

This completes the proof of part (a).

Now from

it follows that

Since (the sign depending on whether n is even or odd), the above inequality implies

This completes the proof of the theorem.

Example 12 Estimate the interval in which the limit of the converging series lies.

Solution

The given is the alternating harmonic series, which is convergent as seen in Example 3. Let the series converges to the real number L.

Then and since , by Theorem 3, for any natural number

Putting n = 9, (i.e., if we truncate the series after the ninth term), we get

i.e., .

Hence the limit of the given alternating series lies in the interval .

Summary

In this session we have discussed the concept of alternating series. We have described Leibniz test for verifying the convergence of alternating series and used it to examine the convergence of some alternating series. A theorem on approximating sums of alternating series have been discussed.

Assignments

Which of the alternating series in Exercises 1-5 converge, and which diverge? Give reasons for your answers.

6. Prove that diverges.

7. Show that diverges.

FAQ

1. State non decreasing Sequence Theorem.

Answer. Non decreasing Sequence Theorem states that nondecreasing sequence of real numbers converges if and only if it is bounded from above. If a nondecreasing sequence converges, it converges to its least upper bound.

2. Series with positive terms converge or diverge to infinity, while alternating series either converge or oscillate. Comment.

Answer: If a series has only positive terms, the partial sums get larger and larger. If they get large too rapidly, the series will diverge. However, if some of the terms are negative, the negative terms may cancel with the positive terms and prevent the partial sums from blowing up.

3. How to determine whether an alternating series converges?

Answer. An alternating series is a series where the terms alternate between positive and negative. We can say that an alternating series converges if the following two conditions are satisfied:

· Its terms are non-increasing — in other words, each term is either smaller than or the same as its predecessor (ignoring the minus signs).

· Its nth term converges to zero.

Using this simple test, we can easily show many alternating series to be convergent. The terms just have to converge to zero and get smaller and smaller (they rarely stay the same). The alternating harmonic series converges by this test.

4. Can you say that if the conditions of Leibniz Test (Alternating Series Test) do not hold, then the alternating series diverge?

Answer:

The statement of Leibniz Test is as follows: Suppose is a sequence of positive numbers such that

(a) (i.e., the ’ s decrease) and

(b) ,

Then both the alternating series and are convergent.

If the first condition (i.e., the terms decrease) of the Leibniz Test doesn’t hold, we cannot say that the alternating series diverges. The Leibniz Test only says that if the conditions are true, then the series converges; it does not say what happens if the conditions are not true. However, if second condition doesn’t hold, i.e., if , we can conclude that the series diverges, by the Zero Limit Test of divergence of series.

5. State Zero Limit Test of divergence of series.

If , the series

diverges.

6. Can the Zero Limit Test of divergence of series can be used to test the divergence of alternating series?

Answer:

If , we can conclude, by the Zero Limit Test of divergence of series, that the alternating series of the forms and both diverge.

QUIZ

1. Which of the following series is not convergent?

(a)

(b)

(c)

(d) none of the above

Ans. (a)

2. If the the first condition (i.e., the terms decrease) of the Leibniz Test is not satisfied by an alternating series, then which of the following is true.

(a) the alternating series is convergent

(b) the alternating series is divergent

(d) none of the above

Ans. (d)

3. A sequence converges to a limit L if and only if ______

(a) the sequence of even-numbered terms converges to L.

(b) the sequence of odd-numbered terms converges to L.

(d) none of the above

Ans. (c)

4. If an alternating series satisfies the hypotheses of the Leibniz test, and if L is the sum of the series, then which of the following is true.

(a)

(b)

(c)

(d) none of the above

Ans. (a)

Glossary

· Series: If be a sequence of real numbers, then

is called a series of real numbers. is the nth term of the series. A series of non-negative terms is the series of real numbers with all terms non-negative.

· nth partial sum: The sum of the first n terms of a series is called the nth partial sum of the series

· Sequence of nth partial sums: The sequence