Syllabus | B.Sc (Non- Medical) | Title- Sequences and Series (09010401)

Title- Sequences and Series (09010401)

S.No.

Unit

Content

Remarks

Time
(Hours)

1

One

  • 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

Nice to know

12

2

Two

  • 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
  • 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

22

3

Three

  • 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
  • Pringshiem’s theorem(statement only)
  • Multiplication of series, Cauchy product of series(definition and examples only)
  • Convergence and absolute convergence of infinite products

Must know

Nice to know

10

Books Recommended:

  • R.R. Goldberg: Real Analysis, Oxford & I.B.H. Publishing Co., New Delhi 1970
  • S.C. Malik: Mathematical Analysis, Wiley Eastern Ltd., Allahabad.
  • Shanti Narayan: A course in Mathematical Analysis, S. Chand and company, New Delhi.
  • Murray, R. Spiegel: Theory and Problems of Advanced Calculus, Schaum Publishing Co., New York
  • T.M. Apostol: Mathematical Analysis, Narosa Publishing House, New Delhi.
  • Earl D. Rainville: Infinite series, The Macmillan Co., New York.
Admission Open- 2018