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
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