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A Basic Course in Real Analysis

Ajit Kumar and S. Kumaresan
Publisher: 
Chapman & Hall/CRC
Publication Date: 
2014
Number of Pages: 
302
Format: 
Hardcover
Price: 
89.95
ISBN: 
9781482216370
Category: 
Textbook
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Real Number System
Algebra of the Real Number System
Upper and Lower Bounds
LUB Property and Its Applications
Absolute Value and Triangle Inequality

Sequences and Their Convergence
Sequences and Their Convergence
Cauchy Sequences
Monotone Sequences
Sandwich Lemma
Some Important Limits
Sequences Diverging to
Subsequences
Sequences Defined Recursively

Continuity
Continuous Functions
Definition of Continuity
Intermediate Value Theorem .
Extreme Value Theorem
Monotone Functions
Limits
Uniform Continuity
Continuous Extensions

Differentiation
Differentiability of Functions
Mean Value Theorems
L'Hospital's Rules
Higher-order Derivatives
Taylor's Theorem
Convex Functions
Cauchy's Form of the Remainder

Infinite Series
Convergence of an Infinite Series
Abel's Summation by Parts
Rearrangements of an Infinite Series
Cauchy Product of Two Infinite Series

Riemann Integration
Darboux Integrability
Properties of the Integral
Fundamental Theorems of Calculus
Mean Value Theorems for Integrals
Integral Form of the Remainder
Riemann's Original Definition
Sum of an Infinite Series as an Integral
Logarithmic and Exponential Functions
Improper Riemann Integrals

Sequences and Series of Functions
Pointwise Convergence
Uniform Convergence
Consequences of Uniform Convergence
Series of Functions
Power Series
Taylor Series of a Smooth Function
Binomial Series
Weierstrass Approximation Theorem

A Quantifiers
B Limit Inferior and Limit Superior
C Topics for Student Seminars
D Hints for Selected Exercises
Bibliography
Index

 

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