Table Of ContentMATHEMATICS
HIGHER LEVEL
OPTION 10
SERIES & DIFFERENTIAL EQUATIONS
PREFACE
This book has been specifically written to meet the demands of the Mathematics Higher Level
Option 10: Series and Differential Equations. It presents an extensive and comprehensive
coverage of the syllabus and in some sections goes beyond what is required of the student.
However, this is for the teacher to decide how best to use these sections with their students.
The text has been written in a conversational style so that students will find that they are not
simply making reference to an encyclopedia filled with mathematical facts, but rather find that
they are in some way participating in or listening in on a discussion of the subject matter.
There is an extensive set of examples for students to refer to when learning the subject matter for
the first time or during their revision period. There is an abundance of well-graded exercises for
students to hone their skills. Of course, it is not expected that any one student work through every
question in this text - such a task would be quite a feat. Again it is hoped then that teachers will
guide the students as to which questions to attempt. The questions serve to develop routine skills,
reinforce concepts introduced in the topic and develop skills in making appropriate use of the
graphics calculator.
Throughout the text the subject matter is presented using graphical, numerical, algebraic and
verbal means whenever appropriate. Classical approaches have been judiciously combined with
modern approaches reflecting new technology - in particular the use of the graphics calculator.
A detailed solution manual has also been included as part of this text book. The solution manual
has been made available on a disc, which can be found on the inside of the back cover. To open
the files you will need to have Adobe® Acrobat® loaded on your system as well as Microsoft
Excel® & Microsoft Word®.
As always, we welcome and encourage teachers and students to contact IBID Press with
feedback, not only on their likes and dislikes but suggestions on how the book can be improved as
well as where errors and misprints occur. There will be updates on the IBID Press website in
relation to errors that have been located in the book as well as other relevant material for this
option (for example, alternative solutions to those presented in the solution manual) - so we
suggest that you visit the IBID website at www.ibid.com.au. If you believe you have located an
error or misprint please email Fabio Cirrito at [email protected].
Fabio Cirrito, September 2006
iii
MATHEMATICS — HL (Option): Series and Differential Equations
CONTENTS .-
1 SEQUENCES AND LIMITS 1
1.1 Sequences 1
1.1.1 Introduction 2
1.1.2 Visualising sequences — Graphs of sequences 6
1.2 Limits of Sequences 8
1.2.1 Introduction 8
122 Defining the limit of a sequence 11
123 More limit definitions 18
124 Algebra of limits 22
13 Limit of a Function 27
13.1 Sequences as real-valued functions 27
132 L’Hospital’s Rule — Limits of indeterminate forms 30
14 Improper Integrals 41
141 What are improper integrals? 41
142 Evaluating improper integrals 42
143 Comparison test for improper integrals 45
2 SERIES 51
2.1 Series 51
2.1.1 Introduction 51
2.1.2 Sum of a series 52
2.1.3 Criteria for convergence of a series 53
2.14 Necessary condition for series convergence 53
2.15 An important mathematical curiosity: The number e 56
2.16 The sufficient condition for series convergence 57
Test 1 — The limit comparison test 57
Test 2 — D’ Alembert’s criterion (Ratio test) 59
Test 3 — The Integral Test 66
2.2 More Use of Integrals 71
2.2.1 More on comparing infinite series and their integral counterpart 71
222 Errors in approximating an infinite series 75
23 Which Test to Use? 79
3 ALTERNATING SERIES AND CONVERGENCE RADIUS 83
31 Alternating Series 83
3.1.1 Introduction 83
32 Other Types of Convergence 86
32.1 Conditional and absolute convergence 86
iv
33 Power Series 89
331 Power series and radius of convergence 89
332 Convergence interval of a power series 90
333 Radius of convergence or interval of convergence of a power series? 96
334 Power series in (x — k) where k = 0 99
34 Calculus with Power Series 102
341 Power series as a function 102
342 Differentiating and integrating power series 104
343 Use of power series to solve some tricky integrals 107
SERIES EXPANSION: TAYLOR AND MACLAURIN SERIES "3
4.1 Series Expansion 113
4.1.1 Introduction — Polynomials 113
4.12 Coefficients of a polynomial written in terms of its derivatives 114
413 Advancing to approximating a function by a polynomial 116
414 From the polynomial approximation to the expressions of a function
as an infinite series. Taylor and Maclaurin series 119
4.15 Trigonometric functions 122
4.16 More Maclaurin expansions 124
4.1.7 Generalising the Maclaurin series for a value of a: the Taylor series 125
4138 Summary 125
4.2 Series Expansion of Combined Expressions 132
42.1 Expansion of composite functions 132
422 Taylor expansion involving multiplication 133
43 Applications — Revisited 136
43.1 Two definite integrals 136
DIFFERENTIAL EQUATIONS 139
51 Introduction 139
5.1.1 Reviewing definitions 139
52 Geometrical Methods 141
521 The slope field 141
53 Numerical Methods 146
531 Euler’s method 146
54 Two New First Order d.ess | 156
54.1 Homogeneous first order differential equations 156
542 Solving differential equations using the integrating factor 159
ANSWERS 167
SOLUTION MANUAL - see disc
MATHEMATICS - HL (Option): Series and Differential Equations
Dedication
I dedicate this book to my dearest wife Manola and to my beloved children
Aram and Igor. The three of them are the sunshine of my life and my source
of inspiration. They have also been a source of support and dedication during
the difficult times of my illness and my present recovery. There are no words
to fully convey what this has meant to me.
Eduardo Balanovski
vi
Sequences and Limits - CHAPTER ¥}
1.1.1 INTRODUCTION
12 3
Consider the following sets (a) { N l5 en }
2’ wi’4’ W
® {2,5,9,25,42,...}.
-
While both sets in (a) and (b) represent a sequence of numbers, set (a) gives the impression that —
—
an ‘obvious’ pattern exists. That is, it seems to represent a well defined sequence of numbers. We —
—
say this because each number in this set can be defined by a rule that not only generates that
number but also tells us the position of that number in the sequence.
-
¥
. The first number in set (a), which we call the first term, is % . To indicate this we use the following
notation; u, = % , with the subscript ‘1’ indicating that it is the first term. The second term, = is
. .. . . 3 .
written as #, 2 = N= . Similarly,y the third term is ¥, 3% = 3= and so on. In fact, close observation leads
W
us to deduce that the general term, called the nth term — denoted by «,, , is given by
n
un = m,Wheren = 1,2,3, e
r
u, = n , where n €7 +
n+1
The domain of this sequence is given by the values of n —i.e., the set of positive integers, while
the range (i.e., the numbers in the sequence) is given by the set of rational numbers. Of course, )
the range will depend on the rule, «,, . The three dots at the end of the sequence mean ‘and so on’. \
That is, we are referring to an infinite sequence — which is what we will be dealing with in this
book. Because of thi-s , the sequence {u,, u,, 3, ...} i. s someti. mes writ. ten as {u,} o _, . !
We are now in the position to have the following definition:
In this course we will only be interested in sequences where u, = f(n),ie., the sequence is
given by a well defined function.
MATHEMATICS - HL (Option): Series and Differential Equations
EXAMPI.Ei
The nth term of the sequence is given by u, = % .
The first term, when n =1, is given by u, = % = % .
; . 2 2 1
The second term, when n =2, 1s given by u, = E = 3 = 3
. .. 3 3
The third term, when n =3, is given by u; = 3 = 3
The fourth term, when n= 4, is given by u, = 4_4_1
’ ’ $T 2T 16 4
1131
Therefore the first four terms of the sequence are >34
®) Withu, = n ;l we have, u, = -11—1 =0;
o = 2-1_1,
2 2 2’
e = 3-1 _ 2,
3703 3’
U, = a-1_3
T4 T
123
Therefore the first four terms of the sequence are O, >34
(©) Thistime u, = 2 sothat u;, = 2, uy, = 2, uy = 2, uy; = 2.
The first four terms are 2,2, 2, 2.
That is, each term has a value of 2. In fact, every term in this sequence will be 2 — this is
known as a constant sequence
It is important to make appropriate use of a graphics calculator when working with sequences. To
do this we need to first set the calculator mode to seq (i.e., sequence mode) and Dot (this is for
when we want to plot the sequence). Once that is completed, by pressing the equation editor we
can enter the sequence rule and proceed as required.
Sc i Eng Floti Flotz Flots
0123456789 nMin=1
Degree uin =1
unc Par Pol uinMing=
Conhected glny=
vinhMina=
athi iR =
Horiz G-T winMind=
Sequences and Limits - CHAPTER "}
Using the examples in Example 1.1 we have:
Step 1: Step 2: Step 3:
Ploti Platz Flots TRELE SETU R n_ |utid
rMin=1 ThlStart=1 TR S
UCRIBRA L2 aTbl=11 z £
uCaMin>Ell Indernt: RE Ask :; 3EF
= Derend: [REE Ask £ '.:I.SEZE
- Efi”}é” )= 7 Eags
winMiny= | =1
Step 1: Enter the expression for the sequence.
If you wish, you may also enter the value of u; (ie.,u(nrMin)=0.5)
Step 2: To view the sequence using the Table format you need to first set up the Table
parameters, In this case, set TblStart to 1 (as this represents n = 1) and then increase
the values of n by 1 (i.e., set ATbl =1).
Step 3: View the entries by calling up the TABLE.
It is also possible to generate the sequence by using the following process:
Step 1: Press 2nd STAT (this will bring up the LIST screen).
Step 2: Move cursor to OPS and then select seq(.
Step 3: Enter the expression for the sequence using the following syntax:
seq(expression, variable, term to start from, last term).
Then store the sequence as a list.
Step 1: Step 2: Step 3:
oPS MATH E@E MATH se d(n 2™l s 1
1 t 1885
ile ortD( £.5 .9 375 .25
3Lz dimg Ang+L1
4Si:Llsyg Fs eidl ll ¢ a£1 .5 .5 .373 .25
5! RESID foumSumC
Tlalist(
Using u,, = (-1)"*1 we have: u, = (-1)?
uy, = (-1)3
uy = (-1)*
uy = (1)
Therefore the first four terms of the sequenceare 1,-1,1,-1
(b) Using the graphics calculator we have:
So the first four terms of the sequence are; 1,-2, 3, 4.
MATHEMATICS — HL (Option): Series and Differential Equations
©) Again, making use of the graphics calculator we have:
seqdCl+(( -12"n)n
Givi 325 analsd?
1vugthefimtfimrwnnsas0,35,1. 8 1.5 .GE66566.
2 Ans+L1
EB 1.0 .6666666..
1
i66666666? 1.25%
1
+
2 W
X
= W | I + — |
3
That is, the first four terms are 1, l oo
W?2, i
W
(b) With u, = J/n+4 we have: uy = J1+4 =5, u, = 2+4 = ./6;
uy = J3+4 ug = JA+= 48 .
I = o 2 a.
That is, the first four terms are Jg, JB, J7.48.
. . AT o1 . 21 fi
(¢) Withu, = sin-—- we have: Uy = Sing = 55U = Sine =
2’ 6 2
Uy = sin% =1 and u, = siniy—t = fi
6 2
That is, the first four terms are %, g, 1, —“g .
So far we have only considered sequences whose rules were written explicitly in terms of .
However, it is possible to define sequences implicitly. The process of evaluating successive terms
in these sequences does not require too much more effort. The following examples illustrate this.
