REAL ANALYSIS

Order completeness of Real numbers, open and closed sets, limit point of sets, Bolzano Weirstrass theorem, concept of compactness, Heine Borel theorem.

Real Sequences , Limit and convergence of a sequence, Monotonic sequences, Cauchy’s sequences, Sub sequences and Cauchy’s General principle of convergence,
Infinite series and their convergences – Comparison test, Cauchy’s nth root test, D’Alembert, Raabe’s ,Cauchy’s Test, Logarithmic test.
 

Alternating Series – Leibnitz Test, Absolute and conditional convergence, Properties of continuous function on closed interval, derivable functions:-Derivative of composite function, The inverse function theorem and darboux theorem.

Reimann Integration, Lower and upper Reimann integrals, Properties of Reimann integration, Mean value theorem of Integral calculus, Fundamental theorem of integral calculus.

Uniform convergence , Sequence and series of function – pointwise and uniform convergence , Weirstrass M- Test, Abel’s and Drichlet’s Test for uniform convergence of series of functions . Continuity of the sum functions of the limit fuctions.