New Chapter 10 Infinite Sequences and Series

CHAPTER 10 INFINITE SEQUENCES AND SERIES

In this chapter we shall study sequences and (infinite) series. Even though, this chapter deals

almost exclusively with series, we need to understand some of the basics of sequences in order

to deal with series.

Series plays an important role in the field of ordinary differential equations and without series

large portions of the field of partial differential equations would not be possible. In other

words, series is an important topic even if you so not see any of the applications.

10.1 Sequences

Objectives

Definition of a sequence

Limit of a sequence

A sequence is a list of real numbers written in a particular order. A sequence may or may not

have infinite number of terms. In this chapter we would be dealing with infinite sequence only.

Since sequence is an unending succession of numbers, in general, it is denoted by

Comments:

(i) The dots are used to suggest that the sequence continuous indefinitely, following the same

pattern.

(ii) The numbers in a sequence are called the terms of the sequence. The terms are described

according to the positions they occupy.

(iii) The integer is called the index of , and indicates where occurs in the list.

Example 10.1.1: Write down the first few terms of each of the following sequences.

(a)

M A T 1 3 3 C a l c u l u s w i t h A n a l y t i c G e o m e t r y I I Page 54

(b)

A sequence (or infinite sequence) is a function whose domain is the set of positive integers.

There are two ways to represent sequences graphically. One way is to represent sequence as

points on the real line. The other way is to represent sequence as points in the plane where the

horizontal axis is the index number of the term and the vertical axis is its value.

Example 10.1.2: Graph each of the following sequence.

(a)

(b)

The graph in part(a) leads us to an important idea about sequences. Notice that as increases

the terms of the sequence gets closer and closer to zero. We say that zero is the limit (or

sometimes the limiting value) of the sequence and is written as

.

If a sequence has a limit L, we say that the sequence converges to L, i.e., . A

sequence that does not have a finite limit or limit does not exist is said to diverge.

Example 10.1.3: Determine whether the given sequence converges or diverges. If it converges,

find the limit.

(a) ans:


(b) ans:

(c) ans:

(d) ans:


Suppose that the sequences and }b{ n converge to limits and , respectively, and c is

any constant. Then

(i)

(ii)

(iii)

(iv)

(v)

(vi) (if )

Squeeze Theorem for Sequence

Suppose for all for some and then

.

Example 10.1.4: Determine whether the given sequence converges or diverges. If it converges,

find the limit.

(a) ans:

(b) ans:


Given any sequence we have the following.

(i) A sequence is said to be increasing if for every .

(ii) A sequence is decreasing if for every .

(iii) If is an increasing or a decreasing sequence then it called monotonic.

(iv) If there exists a number m such that every n we say the sequence is bounded

below. The number m is sometimes called a lower bound for the sequence.

(v) If there exists a number M such that for every n we say the sequence is

bounded above. The number M is sometimes called an upper bound for the sequence.

(vi) If a sequence is both bounded below and bounded above we say the sequence is

bounded.

Commonly occurring limits

The following six sequences converge to the limits listed below:

a.

b.

c. , x > 0

d. ,

e.

f.

Homework

Exercise 10.1: 27, 29, 31, 37, 39, 41, 43, 45, 49, 51, 53, 55, 57, 59, 67, 69, 77, 87


10.2 Infinite Series

Objectives

Definition of infinite series

Calculate the sum of the series

Geometric series

nth term test for divergence

An infinite series (or simply a series) is the sum of an infinite sequences of numbers

=

Since there are an infinite number of terms in the series, we shall look at the sum of the first

terms

.

This is a finite sum and can be calculated by normal addition. The number is called the nth

partial sum of the series. As gets larger, the partial sums get closer and closer to a limiting

value. This is similar to the idea of the terms of a sequence approaching a limit. The sequence

{ } is called the sequence of partial sums of the series.

If the sequence { } converges to a limit S, then the series is said to converge and S is called the

sum of the series.

=

If the sequence of partial sums of a series diverges then the series is said to diverge. A divergent

series has no sum.

A telescoping series is a series of the form


Example 10.2.1: Determine whether the series converges or diverges. If it converges, find the

sum.

(a)

(b)

(c)

A geometric series is one of the form where a 0 and r is

the ratio.

A geometric series converges if | r | < 1 and diverges if | r | 1. If the series converges the sum

is r1

a

.

Example10.2. 2: Find the sum of convergent series.

(a)

1nn4

1

(b)

0nn5

3

(c)

Properties of infinite series


If = A and = B and k is a real number, then the following series converge to the

indicted sums.

(i) = + = A + B

(ii) = - = A B

(iii) = k = kA (Any number k)

Note as well the to add/subtract series we need to make sure that both have the same initial

value of the index and the new series will also start at this value. But this cannot be extended

to the multiplication of series. This means that

nth term test for divergence.

If fails to exist or is not equal to zero then the series diverges.

Caution: This theorem does not say that converges if . It is possible for a series to

diverge when .

Example 10.2.3: Determine convergence and divergence of the series using the nth term test

for divergence.

(a)

(b)

(c)

(d)


Homework

Exercise 10.2: 7, 9, 11, 13, 27, 29, 31, 33, 35, 37, 41, 45, 55, 59, 63


Series of Nonnegative Terms

Objectives

Integral test

p-series

Direct Comparison Test

Limit Comparison Test

Ratio test

Root test

10.3 The Integral Test

In this section we shall relate a series to an improper integral and determine the convergence

of a series.

Let be a series with positive terms and let f(x) be the function that results when n is

replaced by x in the formula . Suppose f is a decreasing and continuous function on the

interval . then and dx both converge or both diverge.

(a) If converges then also converges.

(b) If diverges then also diverges.

Comments

(1) The lower limit on the improper integral must be the same value that starts the series.

(2) The function does not actually need to be decreasing and positive everywhere in the

interval. The function has to be eventually decreasing and positive, i.e., it is okay if the

function (and hence series terms) increases or is negative for a while, but eventually the

function (series terms) must decrease and be positive for all terms.


Example 10.3.1: Use the integral test to determine the convergence or divergence of the series.

(a) ans: diverges

(b) ans: converges

(c)ans: converges

The next test follows directly from the integral test.

Convergence of p-series

The p-series

= 1 +

converges if p > 1 and diverges if 0 < p 1.

Example 10.3.2: Determine whether each of the following series converges or diverges.

(a) ans: diverges

(b) ans: converges

(c)ans: diverges

Homework

Exercise 10.3: 3, 7, 9, 11, 13, 15, 19, 21, 25, 31, 35, 37


10.4 Comparison Tests

While the integral test is a nice test, it forces us to do improper integrals which are not always

easy and in some cases may be impossible to solve.

Comparison Test

Let and be series with positive terms and suppose

.

(i) If the “bigger series” converges then the “smaller series” also converges.

(ii) If the “smaller series” diverges then the “bigger series” also diverges.

Caution: Just because the smaller of the two series converges does not say anything

about the larger series. The larger series may still diverge. Likewise, just because we know that

the larger of two series diverges we can’t say that the smaller series will also diverge!

Note: The requirement that , and only need to be true eventually. In

other words, if the first few terms are negative or , the comparison test can still be use.

This test works as long as eventually the conditions are satisfied for all sufficiently large .

Example 10.4.1: Determine the convergence and divergence using the comparison test.

(a) ans: converges

(b) ans: converges

(c) ans: diverges

(d) ans: another test


Based on the previous example, we need some other test to help us determine the

convergence of this series. The following test will allow us to determine the convergence of this

series.

Limit Comparison Test

Suppose the terms of the two series and are positive.

(i) If the is finite and positive then both converges or both diverges.

(ii) If the = 0 and converges then converges.

(iii) If the and diverges then diverges.

Example 10.4.2: Determine the convergence and divergence of the series using limit

comparison test.

(a) ans: converge

(b) ans: diverges

(b) ans: converge

Homework

Exercise 10.4: 1, 3, 5, 7, 9, 11, 13, 19, 21, 25, 27, 35, 41

10.5 The Ratio and Root Tests


The Ratio Test

Let be a series with positive terms and suppose

= .

(i) If < 1 then the series converges.

(ii) If > 1 or then the series diverges.

(iii) If = 1 then the series may converge or may diverge.

Note: This test is particularly useful for series that contain factorials.

Example 10.5.1: Using the ratio test, determine the convergence and divergence of the

series.

(a)

(b)

(c)

(d)

The Root Test

Let be a series with positive terms and suppose

.

(i) If < 1 then the series converges.




Example 10.5.2: Determine the convergence and divergence of the series using the root

test.

(a)

(b)

Homework

Exercise 10.5: 3, 5, 7, 11, 17, 19, 21, 23, 25, 31, 33, 35, 39, 41

Strategy for Series

The following is a general set of guidelines to help us determine the convergence of a series.

1. At a quick glance does it look like the terms of the series does not converge to zero in

the limit, i.e., ? If so, use the nth term Test for Divergence. Note that

you should only use this test if a quick glance suggests that the series may not converge

to zero.

2. Is the series a p-series or a geometric series? If so use the fact that p-series converges if

p >1 and a geometric series if r <1. Remember that often some algebraic manipulation is

required to get a geometric series into the correct form.

3. Is the series similar to a p-series or a geometric series? If so, try the Comparison Test.

4. Is the series a rational expression involving only polynomials or polynomials under

radicals (i.e. a fraction involving only polynomials or polynomials under radicals)? If so,

try the Comparison Test and/or the Limit Comparison Test. Remember however, that in

order to use the Comparison Test and the Limit Comparison Test the series terms all

need to be positive.

5. Does the series contain factorials or constants raised to powers involving n? If so, then

the Ratio Test may work. Note that if the series term contains a factorial then the only

test that would work is the Ratio Test.

10.6 Alternating Series, Absolute and Conditional Convergence

Objectives

Define the alternating series

Alternating series test (Leibniz’s Theorem)


Absolute convergence

Conditional convergence

The absolute convergence test

Alternating series are of the following forms:

where ’s are all positive.

The tests that we looked at for the convergence of series required that all the terms in the

series be positive. There are many series out there that have negative terms in them. As such

we need to start looking at tests for these kinds of series.

The Alternating Series Test

An alternating series converges if the following conditions are satisfied:

(i)

(ii) is decreasing, i.e.,

Caution This test will only tell us when a series converges and not if the series diverges. In

the second condition all that we need to show is that the terms of the series, will eventually

decrease. It is possible for the first few terms of a series to increase and be decreasing for all

after some point.

Example 10.6.1: Determine the convergence or divergence of the series.

(a) ans: converges

(b) ans: converges


(c) ans: diverges

(d) ans: diverges

(e) ans: converges

Comments We sometimes need to do some fair amount of work to show that the terms are

decreasing. Do not make the assumption that the terms will be decreasing and let it go at that.

Example 10.6.2: Determine whether each of the following series is absolutely convergent,

conditionally convergent or divergent.

(a) ans: absolutely convergent

(b) ans: conditionally convergent

(c) ans: absolutely convergent

The Ratio Test for Absolute Convergence

Let be series with non-zero terms and suppose


A series is said to converge absolutely if converges.

If converges absolutely then it also converges.

A series which converges but diverges is said to be conditionally convergent.

(i) If < 1 then the series converges absolutely.



Note This test is useful for series that contain factorials.

Example 10.6.3: Using the Ratio Test for Absolute Convergence, determine whether the

series converges absolutely, converges conditionally or diverges.

(a) ans: converges absolutely

(b) ans: diverges

(b) ans: converges conditionally

Homework

Exercise 10.6: 7, 9 13, 15, 17, 21, 25, 29, 31, 33, 35, 37, 39


10.7 Power Series

Objectives

Power series and its convergence

The radius and interval of convergence

Endpoint Convergence

In this section we will consider series whose terms involve variables.

A series of the form

where ... are constants and x is a variable, is called a power series about a or just a

power series.

Note A power series is that it is a function of x. That is different from any other kind of series

that we have looked at to this point. In all the previous sections we only had numbers in the

series and now we are allowing variables to be in the series as well.

The convergence of the power series is still the question. The difference is that the convergence

of the series will now depend upon the values of x that we put into the series. A power series

may converge for some values of x and not for other values of x. If a numerical value is

substituted for x in the power series , then we obtain a series of constants that may

either converge or diverge. This leads us to the following basic problem.

Problem: For what values of x does a given power series, , converge?

Consider the power series . For what values of x does the series converge?

There is a number R such that the power series will converge for, and will diverge

for . This number is called the radius of convergence for the series. Note that the


series may or may not converge if . What happens at these points does not change

the radius of convergence. The interval of all x’s, including the endpoints if needed, for which

the power series converges is called the interval of convergence of the series.

Where the series converges only for x = 0, we define the radius of convergence to be R = 0; and

where the series converges absolutely for all x, the radius of convergence is defined to be

.

Example 10.7.1: Find the radius of convergence and the interval of convergence for each of the

following power series.

(a)

(b)

(c)

(d)

Homework Exercise 10.7: Find the radius and interval of convergence for the following questions.

3, 5, 7, 11, 13, 15, 17, 27

10.8 Taylor and Maclaurin Series

Objectives


Taylor and Maclaurin series

Constructing Taylor and Maclaurin series

Differentiation and Integration of Power Series

In this section we will start to represent functions with power series. Functions that are

infinitely differentiable generate power series called Taylor series. These series can provide

useful polynomial approximation of the generating functions

If f has derivatives of all orders at a, then we define the Taylor Series for f about x = a to be

=

If we use a = 0, i.e., Taylor Series about x = 0 , we call the series Maclaurin Series for f ( x) or,

=

Example 10.8.1: Find the Taylor series for the function about .

Example 10.8.2: Find the Taylor series for the function about .

Example 10.8.3: Find the Maclaurin series for the functions .

Example 10.8.4: Using the series obtained in Example 10.8.3, find the Maclaurin series for the

function:

(a)

(b)

Example 10.8.5: Find the Maclaurin series for the functions .


Many of the applications of series, especially those in the differential equations fields, rely on

the fact that functions can be represented as a series. In these applications it is very difficult, if

not impossible, to find the function itself. However, there are methods of determining the

series representation for the unknown function. There are many applications of series,

unfortunately most of them are beyond the scope of this course. One application of power

series (with the occasional use of Taylor Series) is in the field of Ordinary Differential Equations

when finding Series Solutions to Differential Equations. Another application of series arises in

the study of Partial Differential Equations. One of the more commonly used methods in that

subject makes use of Fourier Series.

While the differential equations applications are beyond the scope of this course there are

some applications from a Calculus setting that we can look at.

Example 10.8.6: Use the series obtained in Example 10.8.5,

(a) to approximate to 3 decimal places.

(b) to evaluate .

HomeworkExercise 10.8: 7, 9, 13, 15, 19

Exercise 10.9: 13, 17, 23

Exercise 10.10: 29, 31, 33, 35


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