TSLAS (Real Analysis)

Real Analysis

L T P Cr

4 0 0 4.0

Course Objectives: The course will develop a deep and rigorous understanding of real line and of defining terms to prove the results about convergence and divergence of sequences and series of real numbers. These concepts has wide range of applications in real life scenario.

Real Number System ℝ and its properties:

Algebraic and order properties of ℝ, Absolute value of a real number; Bounded above and bounded below sets, Supremum and infimum of a nonempty subset of ℝ, The completeness property of ℝ, Archimedean property, Density of rational numbers in ℝ, Definition and types of intervals, Nested intervals property, Neighbourhood of a point in ℝ, Open and closed sets in ℝ.

Sequences in ℝ: Convergent sequence, Limit of a sequence, Bounded sequence, Limit theorems, Monotone sequences, Monotone convergence theorem, divergent sequence, Subsequences, Bolzano-Weierstrass theorem for sequences, Limit superior and limit inferior for bounded sequence, Cauchy sequence, Cauchy’s convergence criterion.

Infinite Series: Convergence and divergence of infinite series of real numbers, Necessary condition for convergence, Cauchy criterion for convergence; Tests for convergence of positive term series: Integral test, comparison test, D’Alembert’s ratio test, Cauchy’s nth root test; Alternating series, Leibniz test, Absolute and conditional convergence.

Recommended Books:

1. R.G. Bartle and D. R. Sherbert, Introduction to Real Analysis, 3rd Ed., John Wiley and Sons (Asia) Pvt. Ltd., Singapore, 2002.

2. T. M. Apostol, Mathematical Analysis, 2nd Edition, Narosa Publishing House, Reprint 2002

3. S. C. Malik and Savita Arora, Mathematical Analysis, (2nd ed.). New Age International.

4. R.R. Goldberg : Method of Real Analysis, Oxford & I.B.H. Publishing Co., New Delhi, 1970.