REAL ANALYSIS
B.Sc. THIRD YEAR MATHEMATICS
SEMESTER – VI, PAPER -7
SYLLABUS
B.A./B.Sc. SECOND YEAR MATHEMATICS SYLLABUS
SEMESTER – IV, PAPER- 4
REAL ANALYSIS
UNIT – I (12 hrs) : REAL NUMBERS :
The algebraic and order properties of R, Absolute value and Real line, Completeness property of R, Applications of supreme 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 – Cauchey’s general principle of convergence theorem.
UNIT –II (12 hrs) : INFINITIE SERIES :
Series : Introduction to series, convergence of series. Cauchey’s general principle of convergence for series tests for convergence of series, Series of Non-Negative Terms.
1. P-test
2. Cauchey’s nth root test or Root Test.
3. D’-Alemberts’ Test or Ratio Test.
4. Alternating Series – Leibnitz Test.
Absolute convergence and conditional convergence, semi convergence.
UNIT – III (12 hrs) : 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 (12 hrs) : 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, Mean value Theorems; Role’s Theorem, Lagrange’s Theorem, Cauchhy’s Mean value Theorem
UNIT – V (12 hrs) : 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 as the limit of a sum, Mean value Theorems.
Reference Books :
1. Real Analysis by Rabert & Bartely and .D.R. Sherbart, Published by John Wiley.
2. A Text Book of B.Sc Mathematics by B.V.S.S. Sarma and others, Published by S. Chand & Company
Pvt. Ltd., New Delhi.
3. Elements of Real Analysis as per UGC Syllabus by Shanthi Narayan and Dr. M.D. Raisingkania
Published by S. Chand & Company Pvt. Ltd., New Delhi.
Suggested Activities :
Seminar/ Quiz/ Assignments/ Project on Real Analysis and its applications
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