Riemann Integration and Series of Functions

By Sanjay P. K.   |   National Institute of Technology, Calicut
Learners enrolled: 648
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 : Ongoing Course Type : Core Duration : 15 weeks Start Date : 16 Jan 2023 End Date : 12 Apr 2023 Exam Date : Enrollment Ends : 15 Mar 2023 Category : Mathematics Credit Points : 4 Level : Undergraduate

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
Day 2 Module 5: Riemann’s definition of integrability
Day 3 Module 6: Equivalence of definitions.
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

3 Day 1 Module 7: Riemann integral as a sequential limit
Day 2 Module 8: Riemann integrability of monotone functions and continuous functions
Day 3 Module 9: Further examples of Riemann integral of functions
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

4 Day 1 Module 10: Algebraic properties of Riemann integral
Day 2 Module 11: Monotonicity and additivity properties of Riemann integral
Day 3 Module 12: Approximation by step functions
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

5 Day 1 Module 13: Mean value theorem for integrals
Day 2 Module 14: Fundamental Theorem of Calculus (first form)
Day 3 Module 15: Fundamental Theorem of Calculus (second form)
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

6 Day 1 Module 16: Improper integrals of Type-1.
Day 2 Module 17: Improper integrals of Type-2 and mixed type.
Day 3 Module 18: Gamma and beta functions
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

7 Day 1 Module 19: Pointwise convergence of a sequence of functions
Day 2 Module 20: Uniform convergence
Day 3 Module 21: Uniform norm
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

8 Day 1 Module 22: Cauchy criterion for uniform convergence
Day 2 Module 23: Uniform converegnce and continuity
Day 3 Module 24: Uniform convergence and integration
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

9 Day 1 Module 25: Uniform convergence and differentiation
Day 2 Module 26: Review of infinite series
Day 3 Module 27: Absolute convergence
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

10 Day 1 Module 28: Infinite series of functions
Day 2 Module 29: Weierstrass M-test
Day 3 Module 30: Theorems on the continuity and differentiability of the sum function of a series of functions;
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

11 Day 1 Module 31: Limit superior and limit inferior of a numerical sequence.
Day 2 Module 32: Limit inferior, limit superior and convergence
Day 3 Module 33: Properties of limit superior and Limit inferior
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

12 Day 1 Module 34: Power series and its radius of convergence
Day 2 Module 35: Convergence of power series
Day 3 Module 36: Differentiation and integration of power series
Day 4 Interaction based on the three modules covered
Day 5 Subjective Assignment

13 Day 1 Module 37: Convergence of a power series at the endpoints, Abel's Theorem.
Day 2 Module 38: Weierstrass approximation Theorem
Day 3 Module 39: Proof of Weierstrass approximation theorem
Day 4 Interaction based on the three modules covered
Day 5 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.