Learn · A Basic Course in Real Analysis
Lecture 16 of 46Some important limits, Ratio tests for sequences of Real Numbers
Some Important limits, Ratio tests for sequences of Real Numbers
Prof. P.D. Srivastava · IIT Kharagpur
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- A Basic Course in Real Analysis, IIT Kharagpur (at NPTEL)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
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All the lectures
- Rational Numbers and Rational Cuts
- Irrational numbers, Dedekind's Theorem
- Continuum and Exercises
- Continuum and Exercises (Contd.)
- Cantor's Theory of Irrational Numbers
- Cantor's Theory of Irrational Numbers (Contd.)
- Equivalence of Dedekind and Cantor's Theory
- Finite, Infinite, Countable and Uncountable Sets of Real Numbers
- Types of Sets with Examples, Metric Space
- Various properties of open set, closure of a set
- Ordered set, Least upper bound, greatest lower bound of a set
- Compact Sets and its properties
- Weiersstrass Theorem, Heine Borel Theorem, Connected set
- Tutorial - II
- Concept of limit of a sequence
- Some Important limits, Ratio tests for sequences of Real Numbers
- Cauchy theorems on limit of sequences with examples
- Fundamental theorems on limits, Bolzano-Weiersstrass Theorem
- Theorems on Convergent and divergent sequences
- Cauchy sequence and its properties
- Infinite series of real numbers
- Comparison tests for series, Absolutely convergent and Conditional convergent series
- Tests for absolutely convergent series
- Raabe's test, limit of functions, Cluster point
- Some results on limit of functions
- Limit Theorems for functions
- Extension of limit concept (one sided limits)
- Continuity of Functions
- Properties of Continuous Functions
- Boundedness Theorem, Max-Min Theorem and Bolzano's theorem
- Uniform Continuity and Absolute Continuity
- Types of Discontinuities, Continuity and Compactness
- Continuity and Compactness (Contd.), Connectedness
- Differentiability of real valued function, Mean Value Theorem
- Mean Value Theorem (Contd.)
- Application of MVT , Darboux Theorem, L Hospital Rule
- L'Hospital Rule and Taylor's Theorem
- Tutorial - III
- Riemann/Riemann Stieltjes Integral
- Existence of Reimann Stieltjes Integral
- Properties of Reimann Stieltjes Integral
- Properties of Reimann Stieltjes Integral (Contd.)
- Definite and Indefinite Integral
- Fundamental Theorems of Integral Calculus
- Improper Integrals
- Convergence Test for Improper Integrals