|Course Archives Theoretical Statistics and Mathematics Unit|
Course: Real Analysis I
Time: Currently not offered
Syllabus: The language of sets and functions - countable and uncountable sets. Real numbers - least upper bounds and greatest lower bounds. Sequences - limit points of a sequence, convergent sequences; bounded and monotone sequences, the limit superior and limit inferior of a sequence. Cauchy sequences and the completeness of R. Series - convergence and divergence of series, absolute and conditional convergence. Riemanns rearrangement theorem. Various tests for convergence of series. (Integral test to be postponed till after Riemann integration is introduced in Analysis II.) Connection between infinite series and decimal expansions, ternary, binary expansions of real numbers. Cauchy product, Infinite products.
calculus of a single variable - continuity; attainment of supremum and infimum of a continuous function on a closed bounded interval, uniform continuity. Differentiability of functions. Chain Rule, Rolles theorem and mean value theorem. Higher derivatives, Leibniz formula, maxima and minima. Taylors theorem - various forms of remainder, infinite Taylor expansions. LHospital Rule
(a) T. M. Apostol: Mathematical Analysis.
(b) T. M. Apostol: Calculus.
(c) S. Dineen: Multivariate Calculus and Geometry.
(d) . R. R. Goldberg: Methods of Real Analysis.
(e) T. Tao: Analysis I & II.
(f) Bartle and Sherbert: Introduction to Real Analysis.
(g) H. Royden: Real Analysis.
(h) K. A. Ross: Elementary Analysis.
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