MCF1B Real Analysis 1 Scanned Lecture Notes - Anitha Edition

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Real Analysis 1 Premium (Units 1,2 ,3 and 4) Handwritten Premium Lecture Notes, Prepared by Anitha. Specially for M.Sc Candidates. Syllabus Covered based on University of Madras , M.Sc First Semester.

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Syllabus :
REAL ANALYSIS

UNIT-1
FUNCTION OF BOUNDED VARIATION
UNIT-2
RIEMANN STIEILTJES INTEGRAL
UNIT-3
INTEGRATORS OF BOUNDED VARIATION
UNIT-4
DOUBLE SEQUENCE

Content :
UNIT-1

FUNCTION OF BOUNDED VARIATION
Properties of monotonic function
Total variation
Absolute and conditional convergence
Cauchy sequence
Absolute convergence
Conditional convergence
Dirichelets theorem test
Able’s test
Rearrangement of series
Riemann’s theorem on conditionary
Convergent series

UNIT-2

RIEMANN STIEILTJES INTEGRAL
Introduction and notation
Linear property
Change of variable in a Riemann stieltjes integral
Reduction to a Riemann integral
Step function as integrators
Reduction of a Riemann-stieltjes integral to a finite sum
Eulers summation formula
Condition
Camparison theorem

UNIT-3

INTEGRATORS OF BOUNDED VARIATION
Sufficient condition for existence of Riemann-stieltjes integral
Necessary condition for the existence of Riemann-steltjes integrals
Mean value theorem for Riemann stiettjes integral
Second fundamental theorem of integral calculas
Differentiation under the integral sign
Integrchanging the order of integration
Lebegue’s criterion for existence of Riemann integrals

UNIT-4

DOUBLE SEQUENCE
Double series
Rearrangement of double series
Cauchy product
Mertons theorem
Finite product
Absolute convergence
Multiplication of power series
Taylor’s series

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