**Title-Sequences and Series (Code-09050401)**

**B.Sc. Mathematics (Hons) – 4**^{th} semester

^{th}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:

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