Elements Of Real Analysis (GE) SEM - IV Chapter Wise Notes

 

UNIT - I     Basic Properties of the Set of Real Numbers     Field and order properties of ℝ, basic properties and inequalities of the absolute value of a real number, bounded above and bounded below sets, Suprema and infima, The completeness axiom and the Archimedean property of ℝ.

UNIT-II     Real Sequences     Convergence of a real sequence, Algebra of limits, The squeeze principle and applications, Monotone sequences, Monotone convergence theorem and applications, Cauchy sequences, Cauchy criterion for convergence and applications.

UNIT-III     Infinite Series of Real Numbers     Convergence and divergence of infinite series of real numbers, Necessary condition for convergence, Cauchy criterion for convergence of series, Tests for convergence of positive term series, Applications of the integral test, Comparison tests, D’Alembert’s ratio test, Cauchy’s nth root test, Raabe’s test; Alternating series, Leibniz alternating series test, Absolute and conditional convergence.