Sequences : Definition of a sequence, Bounded and Monotonic sequences, Convergent sequence, Cauchy sequences, Cauchy’s Convergence Criterion, Theorems on limits of sequences. Subsequence, Sequential continuity.
Infinite Series : Definition of a series, Tests of convergence, Comparison tests. Cauchy’s integral test, Ratio tests, condensation test, Raabe’s, Logarithmic, Gauss Test, Cauchy’s root test, Alternating series. Leibnitz’s test. Absolute and conditional convergence.
Weierstrass M-test for uniform convergence of sequences of functions and series of functions, simple applications. Determination of Radius of convergence of Power Series, (All tests without proof. Only applications)
Riemann Integration : Partitions, Upper and Lower integrals, Riemann integrability, Conditions of existence of Riemann integrability of continuous and monotonic functions. Algebra of Integrable functions.
Improper integrals : Definitions, statements of their conditions of existence, Tests of convergence of improper integrals, Beta and Gamma Functions and their convergence, Abel’s and Dirichlet’s tests