Course Outcomes:

After successful completion of this course, the student will be able to

1. Get clear idea about the real numbers and real valued functions.

2. Obtain the skills of analyzing the concepts and applying appropriate methods for testing convergence of a sequence/ series.

3. Test the continuity and differentiability and Riemann integration of a function.

4. Know the geometrical interpretation of mean value theorems.

Course Syllabus:

Contents

**UNIT – I REAL NUMBERS **

The algebraic and order properties of R, Absolute value and Real line, Completeness property of R, Applications of supremum property; intervals. (No question is to be set from this portion).

Real Sequences: Sequences and their limits, Range and Boundedness of Sequences, Limit of a sequence and Convergent sequence. The Cauchy’s criterion, properly divergent sequences, Monotone sequences, Necessary and Sufficient condition for Convergence of Monotone Sequence, Limit Point of Sequence, Subsequences and the Bolzano-weierstrass theorem – Cauchy Sequences – Cauchy’s general principle of convergence theorem.

**UNIT –II INFINITIE SERIES **

Introduction to series, convergence of series. Cauchy’s general principle of convergence for series tests for convergence of series, Series of Non-Negative Terms.

1. P-test

2. Cauchy’s nth root test or Root Test.

3. D’-Alemberts’ Test or Ratio Test.

4. Alternating Series – Leibnitz Test. Absolute convergence and conditional convergence.

**UNIT – III CONTINUITY **

Limits : Real valued Functions, Boundedness of a function, Limits of functions. Some extensions of the limit concept, Infinite Limits. Limits at infinity. (No question is to be set from this portion).Continuous functions : Continuous functions, Combinations of continuous functions, Continuous Functions on intervals, uniform continuity.

**UNIT – IV DIFFERENTIATION AND MEAN VALUE THEOREMS **

The derivability of a function, on an interval, at a point, Derivability and continuity of a function, Graphical meaning of the Derivative, Mean value Theorems; Rolle’s Theorem, Lagrange’s Theorem, Cauchy’s Mean value Theorem

**UNIT – V RIEMANN INTEGRATION**

Riemann Integral, Riemann integral functions, Darboux theorem. Necessary and sufficient condition for R – integrability, Properties of integrable functions, Fundamental theorem of integral calculus, integral asthe limit of a sum, Mean value Theorems.

**Fourth Semester mathematics REAL ANALYSIS Material**

Name of the Unit | Download Link |

UNIT – I REAL NUMBERS | Click Here |

UNIT –II INFINITE SERIES | Click Here |

UNIT – III CONTINUITY | Click Here |

UNIT – IV DIFFERENTIATION AND MEAN VALUE THEOREMS | Click Here |

UNIT – V RIEMANN INTEGRATION | Click Here |

**REAL ANALYSIS QUESTION PAPER 2022 DOWNLOAD** CLICK HERE