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A Basic Course in Real Analysis

A Basic Course in Real Analysis. Instructor: Prof. P.D. Srivastava, Department of Mathematics, IIT Kharagpur. This is an introductory course in real analysis, covering topics: the Dedekind theory of irrational numbers; the Cantor theory of irrational numbers; bounded set and the open, closed and compact, etcetera; continuity of the function; differentiability; the Riemann integration and Riemann stieltjes integral; and the improper integral. (from nptel.ac.in)

 Lecture 23 - Tests for Absolutely Convergent Series

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 Lecture 01 - Rational Numbers and Rational Cuts Lecture 02 - Irrational Numbers, Dedekind's Theorem Lecture 03 - Continuum and Exercises Lecture 04 - Continuum and Exercises (cont.) Lecture 05 - Cantor's Theory of Irrational Numbers Lecture 06 - Cantor's Theory of Irrational Numbers (cont.) Lecture 07 - Equivalence of Dedekind and Cantor's Theory Lecture 08 - Finite, Infinite, Countable and Uncountable Sets of Real Numbers Lecture 09 - Types of Sets with Examples, Metric Space Lecture 10 - Various Properties of Open Set, Closure of a Set Lecture 11 - Ordered Set, Least Upper Bound, Greatest Lower Bound of a Set Lecture 12 - Compact Sets and its Properties Lecture 13 - Weierstrass Theorem, Heine Borel Theorem, Connected Set Lecture 14 - Tutorial II Lecture 15 - Concept of Limit of a Sequence Lecture 16 - Some Important Limits, Ratio Tests for Sequences of Real Numbers Lecture 17 - Cauchy Theorems on Limit of Sequences with Examples Lecture 18 - Fundamental Theorem on Limits, Bolzano-Weierstrass Theorem Lecture 19 - Theorems on Convergent and Divergent Sequences Lecture 20 - Cauchy Sequence and its Properties Lecture 21 - Infinite Series of Real Numbers Lecture 22 - Comparison Tests for Series, Absolutely Convergent and Conditional Convergent Series Lecture 23 - Tests for Absolutely Convergent Series Lecture 24 - Raabe's Test, Limit on Functions, Cluster Point Lecture 25 - Some Results on Limit of Functions Lecture 26 - Limit Theorems for Functions Lecture 27 - Extension of Limit Concept (One Sided Limits) Lecture 28 - Continuity of Functions Lecture 29 - Properties of Continuous Functions Lecture 30 - Boundedness Theorem, Max-Min Theorem, and Bolzano's Theorem Lecture 31 - Uniform Continuity and Absolute Continuity Lecture 32 - Types of Discontinuities, Continuity and Compactness Lecture 33 - Continuity and Compactness (cont.), Connectedness Lecture 34 - Differentiability of Real Valued Function, Mean Value Theorem Lecture 35 - Mean Value Theorem (cont.) Lecture 36 - Application of Mean Value Theorem, Darboux Theorem, L'Hospital's Rule Lecture 37 - L'Hospital's Rule and Taylor's Theorem Lecture 38 - Tutorial III Lecture 39 - Riemann/Riemann-Stieltjes Integral Lecture 40 - Existence of Riemann-Stieltjes Integral Lecture 41 - Properties of Riemann-Stieltjes Integral Lecture 42 - Properties of Riemann-Stieltjes Integral (cont.) Lecture 43 - Definite and Indefinite Integral Lecture 44 - Fundamental Theorems of Integral Calculus Lecture 45 - Improper Integrals Lecture 46 - Convergence Test for Improper Integrals