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
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