Title-Sequences and Series (Code-09050401)
B.Sc. Mathematics (Hons) – 4th semester
| S.No. | Contents of the topics | Learning Objectives | Teaching Guidelines | Methodology | Time
(Hour) |
|
1 |
· Boundedness of the set of Real no.; Least upper bound and Greatest lower bound of a set
· Neighborhoods, interior points and isolated points · Limit points , open sets, closed sets · Interior of a set, Closure of a set in Real numbers Bolzano-Weierstrass theorem · Open covers, Compact sets and Heine-Borel theorem |
Students will be able to understand sets of Real no. with limit points, neighborhood, interior and closure of set | Lecture should be effective so that student can grasp the topics easily | Assignments/ seminars/ Class tests/ Presentations |
12 |
|
2 |
· Sequences: real sequences and their convergence
· Theorem on limits of sequences · Bounded and Monotonic sequences · Cauchy’s sequence, Cauchy general principle of convergence · Subsequences, sub sequential limits · Infinite series: Convergence and divergence of infinite series · Comparison tests of positive term infinite series · Cauchy’s general principle of convergence of series · Convergence and divergence of geometric series, p-series |
Students will be able to understand sequences and their convergence, to study infinite series | Lecture should be effective so that student can grasp the topics easily | Assignments/ seminars/ Class tests/ Presentations |
12 |
|
3 |
· Infinite series: D-Alembert’s ratio test
· Rabbe’s Test, Logarithmic Test, De Morgan and Bertrand’s Test Cauchy nth Root Test, · Gauss Test, Cauchy Integral test, Cauchy’s condensation Test |
Students will be able to understand different test for convergence of infinite series | Lecture should be effective so that student can grasp the topics easily | Assignments/ seminars/ Class tests/ Presentations |
12 |
|
4 |
· Alternating series, Leibnitz’s Test, absolute and conditional convergence
· Arbitrary series: Abel’s lemma, Abel’s Test · Dirichlet’s Test, Insertion and removal of parenthesis · Rearrangement of terms in a series, Dirichlet’s theorem, Riemann’s Re-arrangement theorem · Pringshiem’s theorem(statement only) · Multiplication of series, Cauchy product of series(definition and examples only) · Convergence and absolute convergence of infinite products |
Students will be able to understand Alternating series, Arbitrary series , rearrangement of terms in series and product of series | Lecture should be effective so that student can grasp the topics easily | Assignments/ seminars/ Class tests/ Presentations |
8 |
Books Recommended:
- R. Goldberg: real Analysis, oxford & I.B.H. Publishing Co., New Delhi 1970
- C. Malik: Mathematical Analysis, Wiley Eastern Ltd., Allahabad.
- Shanti Narayan: A course in Mathematical Analysis, S. Chand and company, New Delhi.
- M. Apostol: Mathematical Analysis, Narosa Publishing House, New Delhi.