Syllabus
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Course Code: B-MAT 302 Course Name: Real Analysis -I |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Finite and infinite sets, countable and uncountable sets, bounded and unbounded sets in ℝ. Least upper bound (supremum), greatest lower bound (infimum) of a set and their properties. The set of real numbers (ℝ) as an ordered field, Least upper bound properties of ℝ, Metric property and completeness of ℝ. Archimedean property of ℝ. Neighbourhood of a point, interior points, isolated points, limit points. Open sets, closed sets, interior of a set, closure of a set in real numbers and their properties. Bolzano-Weierstrass theorem. | |
| 2 | Sequences in ℝ, Convergent sequence and its limit, Limit theorems, Bounded and monotonic sequences in ℝ. Cauchy’s theorem on limits, Monotone convergence theorem, Limit superior and limit inferior, Cauchy sequence, Cauchy’s convergence criterion. Subsequences, Subsequential limits. | |
| 3 | Infinite series: Convergence and divergence of Infinite Series, Comparison Tests of positive terms Infinite series, Cauchy’s general principle of Convergence of series, Convergence and divergence of geometric series, Hyper Harmonic series or p-series. D-Alembert’s ratio test, Raabe’s test, Logarithmic test, Abel’s test, Cauchy’s nth root test, Gauss Test, Cauchy’s integral test, Cauchy’s condensation test. | |
| 4 | Alternating series, Absolute and conditional convergence, Leibniz test, Rearrangements of series. Pointwise and uniform convergence of sequence and series of functions, Mn-test, Weierstrass’s M-test. Uniform continuity. Uniform convergence and continuity. | |
