Real Analysis
Good Books for reference:
Introduction to analysis (Bartley and Sherbert)
Syllabus for Real Analysis
Real number system as an ordered field with least upper bound property
Sequences, limit of a sequence, Cauchy sequence, completeness of real line
Lecture 2 What are sequences?
Lecture 3 Behaviour of sequences.
Lecture 4: Epsilon N definition of Limits of sequences
Lecture 5: Recursive definition of sequences.
Lecture 6: What are subsequences?
Lecture 7: Nondecreasing sequences.
Lecture 8: Bounded sequences
Lecture 9: Bounded and Nondecreasing sequences are convergent
Lecture 10: Properties of Limits of sequences
Lecture 11: Frequently arising Limits
Series and its convergence, Absolute and conditional convergence of series of real and complex terms, Rearrangement of series
Lecture 12: Introduction to series.
Lecture 13: Checking the convergence of a series by the sequence of partial sums
Lecture 14: Geometric series
Lecture 15: Some Problems based on geometric series
Lecture 16: Telescopic series
Lecture 17: nth term test for checking the divergence of a series.
Lecture 18: Some noteworthy points regarding convergence of series.
Lecture 19: The integral test for checking the convergence of series of positive terms
Lecture 20: Problems based on integral test.
Lecture 21: p test for checking convergence/divergence of a series
Lecture 22: Comparison tests ( Direct comparison test and limit comparison test ).
Lecture 23: Problems on Limit comparison test (LCT) and Direct comparison test (DCT)
Lecture 24: Ratio test
Lecture 25: Nth root test.
Lecture 26: Alternating series
Continuity and uniform continuity of functions
Properties of continuous functions on compact sets
Riemann integral
Improper integrals
Fundamental theorems of integral calculus
Uniform convergence
Continuity, differentiability and integrability for sequences and series of functions
Partial derivatives of functions of several (two or three) variables
Maxima and minima