Mathematical Analysis
Course Code: MM5CRT01 Coordinator: Dr. Manju K. Menon
Course Credit: 4
Semester: V Class: BSc Mathematics
Course Type: Core
Department: Mathematics
COURSE OBJECTIVES
To make students understand the Real number system and explore the basic terminologies in Real Analysis which is one of the main core subjects in Mathematics.
Learning Outcomes
After the completion of this course the student will be able to
 Determine the basic properties of subsets of the real numbers
 Describe the real line as a complete, ordered field
 Describe about sequences, limit of a sequence and its applications
 Identify the nature of sequences such as convergence, divergence, etc.
 Explain about infinite series and its nature
 Choose tests and analyze the convergence of a series
 Discuss about the absolute convergence of a series
Syllubus
MODULE I: REAL NUMBERS
Finite and Infinite Sets, The Algebraic and Order Properties of R, Absolute Value and Real
Line, The Completeness Property of R, Applications of the Supremum Property, Intervals.
Chapter 1: Section 1.3 and Chapter 2 : Sections 2.1, 2.2,2.3,2.4,2.5
MODULE II: SEQUENCES
Sequences and their Limits, Limit Theorems, Monotone Sequences, Subsequences and the
Bolzano Weierstrass Theorem, The Cauchy Criterion, Properly Divergent Sequences.
Chapter 3 : Sections 3.1,3.2,3.3,3.4, 3.5,3.6
MODULE III: SERIES
Introduction to Series, Absolute Convergence, Tests for Absolute convergence, Tests for
Non absolute Convergence
Chapter 3 : Section 3.7, Chapter 9 : Sections 9.1,9.2,9.3
MODULE IV: LIMITS
Limits of Functions, Limit Theorems, Some Extensions of the Limit Concept
Chapter 4 : Sections 4.1,4.2,4.3
COURSE OUTLINE
No  Topics  Learning Outcomes  Hours 
1 
Finite and Infinite Sets
The Algebraic and Order Properties of R
Absolute Value and Real Line
The Completeness Property of R
Applications of the Supremum Property
Intervals.

· Understands the concept of Subsets of realnumbers and its basic properties · Algebraic Properties of R · Order Properties of R
· Absolute Value · The Properties like OC, of R and i · Supremum, Infimum and its properties · Intervals and Nesterd interval properties · R is countable 
6
2
2
1
3
5
5
3 
Seminar will be given to 1/4 th of students  
2  Sequences and their Limits
Limit Theorems
Monotone Sequences
Subsequences and the Bolzano Weierstrass Theorem
The Cauchy Criterion
Properly Divergent Sequences.

· Have an idea about Sequences and their Limits
· Learns Theorems on limits · Understands Monotone sequences and their properties · Have a clearcut idea about subsequences · Understands Bolzano Weierstrass Theorem · Understands Cauchy Criterion and its uses · Have an idea about Convergence/ divergence of sequences 
3
3
3
2
3
3
5 
Seminar will be given to ¼ th of students  
Internal I  
3  Introduction to Series
Absolute Convergence
Tests for Absolute convergence
Tests for Non absolute Convergence 
· Understands the difference between sequence and series · Understands properties of Series · Differentiate between convergence and absolute convergence and understands whether a series is absolute convergent · Convergence and divergences tests · Given a series, they could test for the convergence/divergence 
1
3
5
10
4 
Seminar will be given to ¼ th of students students  
4  Limits of Functions
Limit Theorems
Some Extensions of the Limit Concept. 
· Understands limit concepts
· Studies theorems on limits · Understands some extensions of limits 
4
4
4

Seminar will be given to ¼ th of students students  
Internal II 
TEACHING SCHEDULE
TOPICS  HOURS  DATES  DESCRIPTION  
Module I


Subsets of Real numbers and its basic properties

6  July 15, 16, 17, 20, 21, 22  
Algebraic Properties of R  2  July 23, 24  
Order Properties of R  2  July 27, 28  
Absolute Value  1  July 29  
The Order Completeness and other Properties of R  3  August 3, 4, 5  
Supremum, Infimum and its properties  5  August 1014  
Intervals and Nesterd interval properties  5  August 1721  
R is countable  3  August 2527  
Discussion on Important Ideas  September 7  
Unit Test  September 8  
Module II


Have an idea about Sequences and their Limits and studies the Theorems 
6 
September 9 11, 1416 

Monotone sequences and their properties  3  September 17, 18, 22  
Subsequences and Bolzano Weierstrass Theorem

5  September 23 25, 2829  
Cauchy Criterion and its uses  3  October 12  
Convergence and Divergence  8  October 514  
Module III


Introduction to Series  4  October 15, 16  Two hrs each  
Absolute Convergence  5  October 1921  
Tests for Absolute convergence  8  October 2226  
Tests for Non absolute Convergence  6  October 2730  
Module IV


Limits of Functions

4  November 24  
Limit Theorems  4  November 57  
Some Extensions of the Limit Concept.  4  November 913  
PEDAGOGY
After the course the students will understand the Basic Concepts in Mathematical Analysis. The tutorials will focus on the learning and understanding of concepts. This course is a stepping stone to study the Real Analysis.
EVALUATION STRATEGY
The internal evaluation is based on the students attendance, seminar, Internal Assessments and assignment marks.
EVALUATION SCHEME
Sl. No  Components  Weightage 
1  END TERM EXAM  80 
2  ATTENDANCE  5 
3  INTERNAL MARKS  10 
4  SEMINARS/ ASSIGNMENT/ VIVA  5 
TEXT BOOK :
Introduction to Real Analysis – Robert G Bartle and Donald R Sherbert (3rd Edition) John Wiley & Sons, In. 2007
REFERENCE BOOKS:
 Richard R Goldberg – Methods of real Analysis, 3rd edition , Oxford and IBM
Publishing Company (1964)
 Shanti Narayan – A Course of Mathematical Analysis, S Chand and Co. Ltd ( 2004)
 Elias Zako – Mathematical Analysis Vol 1, Overseas Press, New Delhi ( 2006)
 J.M Howie – Real Analysis, Springer 2007.
 K.A Ross Elementary – Real Analysis, Springer, Indian Reprints.
 S.C Malik, Savitha Arora – Mathematical Analysis, Revised Second Edition
WEB REFERENCE:
FACULTY DETAILS
Website: stpauls.ac.in
Email: [email protected]
Mobile: 9846335837
SEMINARS/ASSIGNMENTS
Topics will be announced in the online/offline class during the course.