Real Analysis

Unit 1 · Real Numbers

Contributions of Indian mathematicians (Swami Bharati Krishna Tirth, Madhav, Neelkanth Somayaji, Srinivasa Ramanujan). The set of real numbers R as an ordered field, least-upper-bound property, metric property and completeness of R, Archimedean property, dense subsets of R, nested-intervals property, neighbourhood of a point, open sets, limit point of a set, closed and perfect sets in R.

Unit 2 · Convergence of Sequences in R

Bounded and monotonic sequences, convergent sequences and their limits, limit theorems, monotone convergence theorem, subsequences, Bolzano-Weierstrass theorem, limit superior and limit inferior, Cauchy sequences and Cauchy's convergence criterion.

Unit 3 · Infinite Series

Convergence and divergence of infinite series of positive real numbers, necessary condition for convergence, Cauchy criterion; tests for convergence — comparison test, D'Alembert's ratio test, Cauchy root test, Raabe's test, logarithmic test, Cauchy integral test; alternating series and Leibnitz's test, series of arbitrary terms, absolute and conditional convergence, rearrangement of series and Riemann's theorem.

Unit 4 · Riemann Integration & Improper Integrals

Riemann integrability of bounded functions, examples of R-integrable and non-integrable functions, algebra of Riemann-integrable functions, integrability of continuous and monotonic functions, Darboux theorems, the fundamental theorem of integral calculus, and improper integrals.