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Curriculum | Title Sequences and Series (Code 09050401) | B.Sc. Mathematics (Hons) 4th semester

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:

  1. R. Goldberg: real Analysis, oxford & I.B.H. Publishing Co., New Delhi 1970
  2. C. Malik: Mathematical Analysis, Wiley Eastern Ltd., Allahabad.
  3. Shanti Narayan: A course in Mathematical Analysis, S. Chand and company, New Delhi.
  4. M. Apostol: Mathematical Analysis, Narosa Publishing House, New Delhi.