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### Title- Sequences and Series (09010401)

 S.N Unit Time (Hrs) Content of the topics Learning objects Teaching Guidelines Methodology 1 one 15 Boundedness of the set of real numbers; least upper bound, greatest lower bound of a set, neighborhoods, interior points, isolated points, limit points, open sets, closed set, interior of a set,closure of a set in real numbers and their properties. Bolzano-Weiestrass theorem, Open covers,Compact sets and Heine-Borel Theorem. Students will be able to   1.Boundedness of the set of real numbers; least upper bound, greatest lower bound of a set, 2.Bolzano-Weiestrass theorem. 3.Heine-Borel Theorem. Lecture should be effective so that student can grasp the topics easily. 1.  Lecture. 2.  Seminar. 3. Discussion/Interaction with Students. 4. Assignment. 5.  Discussion on Assignment.         6. Evaluation of Assignment. 2 Two 15 Sequence: Real Sequences and their convergence, Theorem on limits of sequence, Bounded and monotonic sequences, Cauchy’s sequence, Cauchy general principle of convergence,   Subsequences, Subsequential limits.   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.   Infinite series: D-Alembert’s ratio test,   Raabe’s test, Logarithmic test, de Morgan and Bertrand’stest, Cauchy’s Nth root test, Gauss Test, Cauchy’s integral test, Cauchy’s condensation test. Students will be able to 1.    Sequence,. Cauchy general principle of convergence, 2.  Hyper Harmonic series or p-series. 3. Raabe’s test, Logarithmic test, de Morgan and Bertrand’stest Lecture should be effective so that student can grasp the topics easily. 1.     Lecture. 2.     Seminar. 3.    Discussion/Interaction with Students. 4.    Assignment. 5.    Discussion on Assignment.         6.    Evaluation of Assignment 3 Three 15 Alternating series, Leibnitz’s test, absolute and conditional convergence, Arbitrary series:   abel’s lemma, Abel’s test, Dirichlet’s test, , Dirichlet’s theorem, Riemann’s Re-arrangement theorem, Pringsheim’s theorem (statement only), Multiplication of series, Cauchy product of series, (definitions and examples only) Convergence and absolute convergence of infinite products. To make students familiar with Alternating series, Leibnitz’s test, absolute and conditional convergence, Arbitrary series: Multiplication of series, Cauchy product of series, (definitions and examples only) Convergence and absolute convergence of infinite products Lecture should be effective sothat student can grasp the topics easily. 1.     Lecture. 2.    Seminar. 3.    Discussion/Interaction with Students. 4.    Assignment. 5.    Discussion on Assignment.        6.    Evaluation of Assignment