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# Riemann Integration and Series of Functions

### Overview

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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'.

### Syllabus

Weeks Weekly Lecture Topics (Module Titles)

1 Day 1 Module 1: Introduction to Riemann integration, Darboux sums. Day 2Module 2: Inequalities for upper and lower Darboux sums. Day 3Module 3: Darboux integral Day 4Interaction based on the three modules covered Day 5Subjective Assignment   2 Day 1 Module 4: Cauchy criterion for integrability Day 2Module 5: Riemann’s definition of integrability Day 3Module 6: Equivalence of definitions. Day 4Interaction based on the three modules covered Day 5Subjective Assignment   3 Day 1 Module 7: Riemann integral as a sequential limit Day 2Module 8: Riemann integrability of monotone functions and continuous functions Day 3Module 9: Further examples of Riemann integral of functions Day 4Interaction based on the three modules covered Day 5Subjective Assignment   4 Day 1 Module 10: Algebraic properties of Riemann integral Day 2Module 11: Monotonicity and additivity properties of Riemann integral Day 3Module 12: Approximation by step functions Day 4Interaction based on the three modules covered Day 5Subjective Assignment   5 Day 1 Module 13: Mean value theorem for integrals Day 2Module 14: Fundamental Theorem of Calculus (first form) Day 3Module 15: Fundamental Theorem of Calculus (second form) Day 4Interaction based on the three modules covered Day 5Subjective Assignment   6 Day 1 Module 16: Improper integrals of Type-1. Day 2Module 17: Improper integrals of Type-2 and mixed type. Day 3Module 18: Gamma and beta functions Day 4Interaction based on the three modules covered Day 5Subjective Assignment   7 Day 1 Module 19: Pointwise convergence of a sequence of functions Day 2Module 20: Uniform convergence Day 3Module 21: Uniform norm Day 4Interaction based on the three modules covered Day 5Subjective Assignment   8 Day 1 Module 22: Cauchy criterion for uniform convergence Day 2Module 23: Uniform converegnce and continuity Day 3Module 24: Uniform convergence and integration Day 4Interaction based on the three modules covered Day 5Subjective Assignment   9 Day 1 Module 25: Uniform convergence and differentiation Day 2Module 26: Review of infinite series Day 3Module 27: Absolute convergence Day 4Interaction based on the three modules covered Day 5Subjective Assignment   10 Day 1 Module 28: Infinite series of functions Day 2Module 29: Weierstrass M-test Day 3Module 30: Theorems on the continuity and differentiability of the sum function of a series of functions; Day 4Interaction based on the three modules covered Day 5Subjective Assignment   11 Day 1 Module 31: Limit superior and limit inferior of a numerical sequence. Day 2Module 32: Limit inferior, limit superior and convergence Day 3Module 33: Properties of limit superior and Limit inferior Day 4Interaction based on the three modules covered Day 5Subjective Assignment   12 Day 1 Module 34: Power series and its radius of convergence Day 2Module 35: Convergence of power series Day 3Module 36: Differentiation and integration of power series Day 4Interaction based on the three modules covered Day 5Subjective Assignment   13 Day 1 Module 37: Convergence of a power series at the endpoints, Abel's Theorem. Day 2Module 38: Weierstrass approximation Theorem Day 3Module 39: Proof of Weierstrass approximation theorem Day 4Interaction based on the three modules covered Day 5Subjective Assignment

Sanjay P. K.

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