REAL ANALYSIS SYLLABUS

SEMESTER –IV – PAPER – IV

CBCS/ SEMESTER SYSTEM (w.e.f. 2020-21 Admitted Batch)

B.A./B.Sc. MATHEMATICS

REAL ANALYSIS SYLLABUS (75 Hours)

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
2. obtain the skills of analyzing the concepts and applying appropriate methods for testing convergence of a sequence/
3. test the continuity and differentiability and Riemann integration of a
4. know the geometrical interpretation of mean value

Course Syllabus:

REAL NUMBERS:

UNIT – I (12 Hours)

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. (No question is to be set from this portion).

INFINTE SERIES:

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 n^th root test or Root
3. D’-Alemberts’ Test or Ratio

UNIT – II (12 Hours)

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.

UNIT – III (12 Hours)

DIFFERENTIATION AND MEAN VALUE THEORMS :

The derivability of a function, on an interval, at a point, Derivability and continuity of a function, Graphical meaning of the Derivative, Problems on Differentiation.

UNIT – IV (12 Hours)

MEAN VALUE THEOREMS:

Mean value Theorems; Rolle’s Theorem, Lagrange’s Theorem, Cauchy’s Mean value Theorem and their applications.

UNIT – V (12 Hours)

RIEMANN INTEGRATION:

Riemann Integral, Riemann integral functions, Darboux theorem(statement only), Necessary and sufficient condition for R – integrability, Properties of integrable functions, Fundamental theorem of integral calculus, First-Mean Value theorem.

Co-Curricular Activities(15 Hours)

Seminar/ Quiz/ Assignments/ Real Analysis and its applications / Problem Solving.

Text Book:

Introduction to Real Analysis by Robert G.Bartle and Donlad R. Sherbert, published by JohnWiley.

Reference Books:

1.A Text Book of B.Sc Mathematics by B.V.S.S. Sarma and others, published by S. Chand & Company Pvt. Ltd., New Delhi.

2. Elements of Real Analysis as per UGC Syllabus by Shanthi Narayan and Dr. M.D. Raisinghania, published by S. Chand & Company Pvt. Ltd., New Delhi.