FOURTH SEMESTER MATHEMATICS
REAL ANALYSIS
UNIT – I
REAL NUMBERS :The algebraic and order properties of R, Absolute value and Real line, Completeness property of R, Applications of supreme property; intervals. (No question is to be set from this portion).Real Sequences:Sequences and their limits, Range and Bounded ness of Sequences, Limit of a sequence and Convergent sequence. The Cauchy’s criterion, properly divergent sequences, Monotone sequences, Necessary and Sufficient condition for Convergence of Monotone Sequence, Limit Point of Sequence, Sub sequences and the Bolzano-weierstrass theorem – Cauchy Sequences – Cauchy’s general principle of convergence theorem.
UNIT –II
INFINITIES SERIES :Series :Introduction to series, convergence of series. Cauchy’s general principle of convergence for series tests for convergence of series, Series of Non-Negative Terms.
1. P-test and Comparison tests
2. Cauchy’s nth root test or Root Test.
3. D’-Alembert’ Test or Ratio Test.
4. Alternating Series – Leibnitz Test. Absolute convergence and conditional convergence.
UNIT – III
CONTINUITY :Limits : Real valued Functions, Boundedness of a function, Limits of functions. Some extensions of the limit concept, Infinite Limits. Limits at infinity. (No question is to be set from this portion).Continuous functions : Continuous functions, Combinations of continuous functions, Continuous Functions on intervals, uniform continuity.
UNIT – IV
DIFFERENTIATION AND MEAN VALUE THEOREMS :The derivability of a function, on an interval, at a point, Derivability and continuity of a function,Graphical meaning of the Derivative, Mean value Theorems; Rolle’s Theorem, Lagrange’s Theorem,Cauchy’s Mean value Theorem
UNIT – V
RIEMANN INTEGRATION :Riemann Integral, Riemann integral functions, Darboux theorem. Necessary and sufficient condition for R – integrability, Properties of integrable functions, Fundamental theorem of integral calculus, integral asthe limit of a sum, Mean value Theorems.