REAL ANALYSIS ( Mathematics Paper-1 ) Book B.Sc 3rd Semester Bihar

Unit

Subject

No. of Lectures

1

Dedekind theory of real numbers, Algebraic and order properties of R, Archimedean Property, Density Theorem, Completeness property of R, Bounded sets, Theorems on Suprema and lnfima.

10

2

Neighbourhood of a point in R, Open and closed sets, Limit points and isolated points of a set, Bolzano-Weierstrass theorem for a set, Derived set, Clouser and Interior of a set.

12

3

Sequence and its convergence, Bounded sequence, Monotone sequences, Subsequences, Limit of a sequence, Limit Theorem, Bolzano-Weierstrass theorem for sequences, Limit superior and limit inferior for bounded sequence, Cauchy sequence, Cauchy’s general principle of convergence.

14

4

Infinite series and their convergence, Cauchy Criterion, Tests for convergence: Comparison test, D’Alembert Ratio Test, Cauchy's root test, Rabbe’s test, Logarithmic test, D’Morgan and Bertrand’s test, Cauchy integral test, Cauchy condensation test, Gauss’s test, Alternating series, Leibnitz test, Absolute and Conditional convergence.

14

Unit

Subject

No. of Lectures

1

Dedekind theory of real numbers, Algebraic and order properties of R, Archimedean Property, Density Theorem, Completeness property of R, Bounded sets, Theorems on Suprema and Infima.

10

2

Sequence and its convergence, Bounded sequence, Monotone sequences, Subsequences, Limit of a sequence, Limit Theorem, Bolzano-Weierstrass theorem for sequences, Cauchy sequence, Cauchy’s general principle of convergence.

10

3

Infinite series and their convergence, Cauchy Criterion, Tests for convergence: Comparison test, D’Alembert Ratio Test, Cauchy’s root test, Rabbe’s test, Logarithmic test, Cauchy integral test, Gauss’s test, Alternating series, Leibnitz test, Absolute and Conditional convergence.

10