# Riemann Integration and Series of Functions

By Sanjay P. K.   |   National Institute of Technology, Calicut
Learners enrolled: 95
The course "Riemann Integration and Series of Functions" is proposed for B.Sc Mathematics or B.Sc. (Hons) Mathematics students. The course content is divided in to 39 modules and the course credit is four. The first part of the course discusses Riemann's theory of integration. It starts with the definition of the Riemann sum, which naturally leads to the notion of integrals, discusses equivalent conditions for the existence of integral and properties of integral and finally proves the 'Fundamental Theorem of Calculus'. The second part of the course is on the sequence and series of functions, where we will look at the significance of 'uniform convergence' to prove the continuity, differentiability and integrability of the limit function of a sequence of functions. Finally, we will define limit superior and limit inferior and discuss results for the special case of 'power series'.
Summary
 Course Status : Upcoming Course Type : Core Duration : 16 weeks Category : Mathematics Credit Points : 5 Level : Undergraduate Start Date : 08 Jul 2024 End Date : 29 Oct 2024 Enrollment Ends : 31 Aug 2024 Exam Date : 07 Dec 2024 IST Exam Shift : Shift 2

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

### Course layout

Weeks Weekly Lecture Topics (Module Titles)

1 Day 1 Module 1: Introduction to Riemann integration, Darboux sums.
Day 2 Module 2: Inequalities for upper and lower Darboux sums.
Day 3 Module 3: Darboux integral
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

2 Day 1 Module 4: Cauchy criterion for integrability - Part 1
Day 2 Module 5: Cauchy criterion for integrability - Part 2
Day 3 Module 6: Riemann’s definition of integrability
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

3 Day 1 Module 7: Equivalence of definitions Part 1
Day 2 Module 8: Equivalence of definitions Part 2
Day 3 Module 9: Riemann integral as a sequential limit
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

4 Day 1 Module 10: Riemann integrability of monotone functions and continuous functions
Day 2 Module 11: Further examples of Riemann integral of functions
Day 3 Module 12: Algebraic properties of Riemann integral
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

5 Day 1 Module 13: Monotonicity and additivity properties of Riemann integral
Day 2 Module 14: Approximation by step functions
Day 3 Module 15: Mean value theorem for integrals
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

6 Day 1 Module 16: Fundamental Theorem of Calculus (first form)
Day 2 Module 17: Fundamental Theorem of Calculus (second form)- Part 1
Day 3 Module 18: Fundamental Theorem of Calculus (second form)- Part 2
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

7 Day 1 Module 19: Improper integrals of Type-1 - Part 1
Day 2 Module 20: Improper integrals of Type-1 - Part 2
Day 3 Module 21: Improper integrals of Type-2 and mixed type.
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

8 Day 1 Module 22: Gamma and beta functions
Day 2 Review of Riemann integration
Day 3 Module 23: Pointwise convergence of a sequence of functions
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

9 Day 1 Module 24: Uniform convergence
Day 2 Module 25: Uniform norm - Part 1
Day 3 Module 26: Uniform norm - Part 2
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

10 Day 1 Module 27: Cauchy criterion for uniform convergence
Day 2 Module 28: Uniform convergence and continuity
Day 3 Module 29: Uniform convergence and integration
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

11 Day 1 Module 30: Uniform convergence and differentiation - Part 1
Day 2 Module 31: Uniform convergence and differentiation - Part 2
Day 3 Module 32: Review of infinite series
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

12 Day 1 Module 33: Absolute convergence - Part 1
Day 2 Module 34: Absolute convergence - Part 2
Day 3 Module 35: Infinite series of functions
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

13 Day 1 Module 36: Weierstrass M-test
Day 2 Module 37: Theorems on the continuity and differentiability of the sum function of a series of functions
Day 3 Module 38: Limit superior and limit inferior of a numerical sequence.
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

14 Day 1 Module 39: Limit inferior, limit superior and convergence
Day 2 Module 40: Properties of limit superior and limit inferior
Day 3 Module 41: Power series and its radius of convergence
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

15 Day 1 Module 42: Convergence of power series
Day 2 Module 43: Differentiation and integration of power series - Part 1
Day 3 Module 44: Differentiation and integration of power series - Part 1
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

16 Day 1 Module 45: Convergence of a power series at the endpoints
Day 2 Module 46: Abel's Theorem
Day 3 Module 47: Real analytic functions
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

17 Day 1 Module 48: Weierstrass approximation Theorem
Day 2 Module 49: Proof of Weierstrass approximation theorem
Day 3 Interaction based on the three modules covered
Day 4 Subjective Assignment

### Books and references

1. K.A. Ross, Elementary Analysis, The Theory of Calculus, Undergraduate Texts in Mathematics, Springer (SIE), Indian reprint, 2004.
2. R.G. Bartle D.R. Sherbert, Introduction to Real Analysis, 3rd Ed., John Wiley and Sons (Asia) Pvt. Ltd., Singapore, 2002.
3. Charles G. Denlinger, Elements of Real Analysis, Jones & Bartlett (Student Edition), 2011.

### Sanjay P. K.

National Institute of Technology, Calicut
Sanjay P. K. has 19 years of experience in teaching both undergraduate level and post graduate level. Taught ‘Real Analysis’, Complex Analysis’, ‘Topology’ and ‘Linear Algebra’ for M.Sc. propgramme at NIT Calicut. Awarded Ph. D.. from the Indian Institute of Science in 2012.
Selected for support under the Mathematical Research Impact Centric Support (MATRICS) scheme of the Science and Engineering Research Board (SERB) in 2018.

### Course certificate

Assessment/Assignment marks will be considered for Internal Marks and will carry 30 percent for overall Result.

End Term Exam- will have 100 questions and will carry 70 percent of overall Result.

*All students, who obtain 40% marks in in-course assessment and 40% marks in end-term proctored exam separately, will be eligible for certificate and credit transfer.