T 2C
MC/EL/CE/RN 260
MATHEMATICAL ANALYSIS
BSc
Compiled
By
HENRY OTOO
(NEW)
JANUARY, 2020
, MATHEMATICAL ANALYSIS
MC/EL/CE/RN 260
INSTRUCTOR: Henry Otoo
Email: /
OFFICE: Mathematics Department / Phone: 0542823234
OFFICE HOURS: 1:00pm - 3:00pm Tuesdays, Wednesdays
THERE WILL BE RANDOM QUIZZES AT ANYTIME WITHIN THE
SEMESTER
OBJECTIVES: In classical mathematical analysis the objects of study (analysis) were
first and foremost functions. "First and foremost" because the development of
mathematical analysis has led to the possibility of studying, by its methods, forms
more complicated than functions: functionals, operators, etc.
Hence the objective importance of mathematical analysis as a means of studying
functions. Nevertheless, the term "mathematical analysis" is often used as a name for
the foundations of mathematical analysis, which unifies the theory of real numbers,
the theory of limits, the theory of series, differential and integral calculus, and their
immediate applications such as the theory of maxima and minima, Fourier Series and
Fourier Integrals.
AIM: Everywhere in nature and technology one meets motions and processes which
are characterized by functions; the laws of natural phenomena also are usually
described by functions.
This course introduces the concepts of convergence of series and sequences. The
course covers the application of convergence of series in power series. The course
introduces Taylor and Maclaurin’s series, Fourier series and evaluate integrals and
factorial with reference to gamma and beta functions.
PREREQUISITES: It is assumed that the student has some background knowledge
in calculus and linear algebra.
Lectures notes prepared by Henry Otoo (Umat Maths Dept.) Page i
,GRADING CRITERIA and EVALUATION PROCEDURES: The grade for the
course will be based on class attendance, group homework, quizzes/ class test and a
final end of term exams.
1. Attendance: All students should make it a point to attend classes. Random
attendance will be taken to constitute 10% of the final grade.
2. Homework: Homework will be assigned on Thursdays and will be due at 8
am the following Monday.
3. Quizzes / Class test: Quizzes based on theoretical techniques worth 30% of
the grade will be given during class. The exam date will be announced one week
in advance.
4. Final End-of-Term Exams: Final exam is worth 60% of the final grade.
ASSESSMENT OF COURSE
Assessment of students
The student’s assessment will be in two forms:
Continuous Assessment [40%] and
End of semester examination [60%]
Assessment of Lecturer
At the end of the course each student will be required to evaluate the course and
the lecturer’s performance by answering a questionnaire specifically prepared
to obtain the views and opinions of the student about the course and lecturer.
Please be sincere and frank.
STUDENT RESPONSIBILITY: It is assumed that each student attends the lectures
and works all the assigned problems. The student is responsible for ALL material
covered in class and any assigned reading. Students may discuss homework problems
but you are responsible for writing your individual answers. If your name does not
appear on the final class roll, then you will not receive a grade for this course.
EXAM POLICY: No makeup for missed work will normally be given, unless
extenuating circumstances occur. Travel plans are not extenuating circumstances.
Acceptable medical excuses must state explicitly that the student should be excused
from class.
Lectures notes prepared by Henry Otoo (Umat Maths Dept.) Page ii
, TABLE OF CONTENT
TABLE OF CONTENT iii
CHAPTER 1 1
CONVERGENCE OF SERIES 1
1.1 Sequence 1
1.2 Series 1
1.2.1 Types of Series 2
1.3 Infinite Series 9
1.3.1 Infinite Geometric Series 9
1.4 Convergence of Series 11
1.4.1 Series of Positive Terms 17
1.5 Test for Convergence 17
1.5.1 Integral Test 17
1.5.2 The P-Series Test 19
1.5.3 Comparison Test 20
1.5.4 Divergent Test 21
1.5.5 Alternating Series Test 22
1.5.6 Absolute Convergence 24
1.5.7 Ratio Test 26
1.5.8 Root Test 28
CHAPTER 2 32
POWER SERIES, TAYLOR SERIES AND MACLUARIN SERIES 32
2.1 Power Series 32
2.2 Taylor and Maclaurin Series 38
CHAPTER 3 44
GAMMA AND BETA FUNCTIONS 44
3.1 Gamma Function 44
3.1.1 Definition 44
3.1.2 Recurrence Relation for Gamma Formula 44
3.2 Beta Function 49
3.2.1 Reduction Formulae 50
3.2.3 Relation between the Gamma and Beta Function 54
3.2.4 Application of Gamma and Beta Functions 56
CHAPTER 4 58
MULTIPLE INTEGRALS 58
4.1 Iterated Integrals 58
Lectures notes prepared by Henry Otoo (Umat Maths Dept.) Page iii
MC/EL/CE/RN 260
MATHEMATICAL ANALYSIS
BSc
Compiled
By
HENRY OTOO
(NEW)
JANUARY, 2020
, MATHEMATICAL ANALYSIS
MC/EL/CE/RN 260
INSTRUCTOR: Henry Otoo
Email: /
OFFICE: Mathematics Department / Phone: 0542823234
OFFICE HOURS: 1:00pm - 3:00pm Tuesdays, Wednesdays
THERE WILL BE RANDOM QUIZZES AT ANYTIME WITHIN THE
SEMESTER
OBJECTIVES: In classical mathematical analysis the objects of study (analysis) were
first and foremost functions. "First and foremost" because the development of
mathematical analysis has led to the possibility of studying, by its methods, forms
more complicated than functions: functionals, operators, etc.
Hence the objective importance of mathematical analysis as a means of studying
functions. Nevertheless, the term "mathematical analysis" is often used as a name for
the foundations of mathematical analysis, which unifies the theory of real numbers,
the theory of limits, the theory of series, differential and integral calculus, and their
immediate applications such as the theory of maxima and minima, Fourier Series and
Fourier Integrals.
AIM: Everywhere in nature and technology one meets motions and processes which
are characterized by functions; the laws of natural phenomena also are usually
described by functions.
This course introduces the concepts of convergence of series and sequences. The
course covers the application of convergence of series in power series. The course
introduces Taylor and Maclaurin’s series, Fourier series and evaluate integrals and
factorial with reference to gamma and beta functions.
PREREQUISITES: It is assumed that the student has some background knowledge
in calculus and linear algebra.
Lectures notes prepared by Henry Otoo (Umat Maths Dept.) Page i
,GRADING CRITERIA and EVALUATION PROCEDURES: The grade for the
course will be based on class attendance, group homework, quizzes/ class test and a
final end of term exams.
1. Attendance: All students should make it a point to attend classes. Random
attendance will be taken to constitute 10% of the final grade.
2. Homework: Homework will be assigned on Thursdays and will be due at 8
am the following Monday.
3. Quizzes / Class test: Quizzes based on theoretical techniques worth 30% of
the grade will be given during class. The exam date will be announced one week
in advance.
4. Final End-of-Term Exams: Final exam is worth 60% of the final grade.
ASSESSMENT OF COURSE
Assessment of students
The student’s assessment will be in two forms:
Continuous Assessment [40%] and
End of semester examination [60%]
Assessment of Lecturer
At the end of the course each student will be required to evaluate the course and
the lecturer’s performance by answering a questionnaire specifically prepared
to obtain the views and opinions of the student about the course and lecturer.
Please be sincere and frank.
STUDENT RESPONSIBILITY: It is assumed that each student attends the lectures
and works all the assigned problems. The student is responsible for ALL material
covered in class and any assigned reading. Students may discuss homework problems
but you are responsible for writing your individual answers. If your name does not
appear on the final class roll, then you will not receive a grade for this course.
EXAM POLICY: No makeup for missed work will normally be given, unless
extenuating circumstances occur. Travel plans are not extenuating circumstances.
Acceptable medical excuses must state explicitly that the student should be excused
from class.
Lectures notes prepared by Henry Otoo (Umat Maths Dept.) Page ii
, TABLE OF CONTENT
TABLE OF CONTENT iii
CHAPTER 1 1
CONVERGENCE OF SERIES 1
1.1 Sequence 1
1.2 Series 1
1.2.1 Types of Series 2
1.3 Infinite Series 9
1.3.1 Infinite Geometric Series 9
1.4 Convergence of Series 11
1.4.1 Series of Positive Terms 17
1.5 Test for Convergence 17
1.5.1 Integral Test 17
1.5.2 The P-Series Test 19
1.5.3 Comparison Test 20
1.5.4 Divergent Test 21
1.5.5 Alternating Series Test 22
1.5.6 Absolute Convergence 24
1.5.7 Ratio Test 26
1.5.8 Root Test 28
CHAPTER 2 32
POWER SERIES, TAYLOR SERIES AND MACLUARIN SERIES 32
2.1 Power Series 32
2.2 Taylor and Maclaurin Series 38
CHAPTER 3 44
GAMMA AND BETA FUNCTIONS 44
3.1 Gamma Function 44
3.1.1 Definition 44
3.1.2 Recurrence Relation for Gamma Formula 44
3.2 Beta Function 49
3.2.1 Reduction Formulae 50
3.2.3 Relation between the Gamma and Beta Function 54
3.2.4 Application of Gamma and Beta Functions 56
CHAPTER 4 58
MULTIPLE INTEGRALS 58
4.1 Iterated Integrals 58
Lectures notes prepared by Henry Otoo (Umat Maths Dept.) Page iii
