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Title-Sequences and Series (Code-09050401)

B.Sc. Mathematics (Hons) – 4th semester

S.No. Topic Domain Hours as per UGC
 

 

 

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 and their properties

·         Bolzano-Weierstrass theorem

·         Open covers, Compact sets and Heine-Borel theorem

Must know 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, subsequential 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

·         Hyper Harmonic series or p-series

Must know 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

Must know 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

Nice to know 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. Murray, R. Spiegel: Theory and Problems of Advanced Calculus, Schaum Publishing Co., New York
  5. M. Apostol: Mathematical Analysis, Narosa Publishing House, New Delhi.
  6. Earl D. Rainville: Infinite series, The Macmillan Co., New York.
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