In these “Real AnalysisNotes PDF”, you will study the deep and rigorous understanding of real line ℝ. and of defining terms to prove the results about convergence and divergence of sequences and series of real numbers. These concepts have wide range of applications in real life scenario.
The topics we will cover will be taken from the following list:
Real Number System ℝ: 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.
Properties of ℝ: The completeness property of ℝ, Archimedean property, Density of rational numbers in ℝ; Definition and types of intervals, Nested intervals property; Neighborhood 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, 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, Basic comparison test, Limit comparison test, D’Alembert’s ratio test, Cauchy’s nth root test; Alternating series, Leibniz test, Absolute and conditional convergence.
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