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चार महीने का बच्चा कैसे बना अरब़पति

चार महीने का बच्चा कैसे बना अरब़पति? जन्म के सिर्फ चार माह बाद यदि कोई बच्चा अरबपति बन जाए तो इसे उसकी किस्मत ही कहेंगे। भारत के एकाग्रह रोहन मूर्ति नाम के बच्चे की किस्मत कुछ इसी प्रकार चमकी है। देश की दूसरी सबसे बड़ी आइटी कम्पनी इंफोसिस के फाउंडर नारायण मूर्ति ने सोमवार अपने चार महीने के पोते एकाग्रह मूर्ति को 240 करोड़ रूपए के शेयरों की हिस्सेदारी का तोहफा देकर उसे शायद देश का सबसे कम उम्र का अरबपति बना दिया है। BSE की फाइलिंग के अनुसार इंफोसिस में अब एकाग्रह रोहन की 15 लाख शेयरों की हिस्सेदारी हो गई है। इसका मतलब अब एकाग्रह रोहन इंफोसिस का 0.04% का हिस्सेदार है। शेयरों के स्थानान्तरण के बाद नारायण मूर्ति के पास कम्पनी के कुल शेयरों का 0.36% हिस्सा बचा है। जिस समय नारायण मूर्ति द्वारा अपने पोते को शेयर देने की खबर बाई उस समय इंफोसिस के शेयरों में गिरावट देखने को मिल रही थी। एकाग्रह रोहन, नारायण मूर्ति तथा सुधा मूर्ति के बेट रोहन मूर्ति और उनकी पत्नि अर्पणा कृष्णन का बेटा है। आपको यह पता होगा कि नोरायण मूर्ति ने अपनी पत्नि सुधा मूर्ति से 10 हजार रूपए उधार लेकर 1981 में इंफोसिस क
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Real Analysis | Mathematics | BSc

Real Analysis

Real Analysis and Theory of Convergence



Authors: Dr. Vimal Saraswat, Dr. Anil Kumar Menaria, Dr. Gajendrapal Singh Rathore

ISBN : 978-81-7906-338-5

Price: Rs. 395.00

Publisher:Himanshu Publications, Hiran Magri Udaipur; Himanshu Publications Prakash House, Ansari Road, New Delhi

E-mail : info@sacademy.co.in

Phone: +91 9664392614

To buy this book click on the link Real Analysis by Saraswat

This book includes the following topics 

Real Number System

  • Introduction
  • Field axiom
  • Uniqueness property
  • Cancellation law of addition and multiplication
  • Order axiom and ordered field
  • Positive class
  • Boundedness
    • Upper bound, Supremum, Lower bound, Infimum, Bounded set
  • Greatest and least element
  • Completeness axiom
  • Complete ordered field
  • Archimedean property of real numbers
    • Archimedean ordered field
  • Betweenness theorem
  • Dedekind's completeness axiom
  • Irrational numbers
  • Rational density theorem or denseness property
  • Absolute value of a real number or Modulus

Point Set Topology

  • Introduction
  • Neighbourhood (nbd) of a real number
  • Properties of neighbourhood
  • Interior and exterior point of a set
  • Interior of a set
  • Open set
  • Limit point of a set
  • Derived set and closed set
  • Closure
  • Open and closed interval
  • Nested interval property
  • Bolzano-Weierstrass theorem
  • Complement of set
  • Open cover, subcover and compact set
  • Heine Borel theorem
  • Connected and disconnected set

Countable Sets

  • Introduction
  • Equivalent sets
  • Finite and infinite set
  • Countable set
  • Uncountable set
  • Cantor ternary set
  • Binary representation
  • Ternary representation
  • Construction of Cantor ternary set
  • Properties of Cantor ternary set

Real Sequences

  • Introduction
  • Sequence
  • Range of a sequence
  • Bounded and unbounded sequence
  • Supremum and infimum of sequence
  • Monotonic sequence
  • Limit point of a sequence
  • Bolzano-Weierstrass theorem
  • Limit of a sequence
  • Convergent sequence
  • Divergent sequence
  • Oscillatory sequence
  • Theorems on convergence sequences
  • Theorems on convergence of monotonic sequences
  • Algebra of sequences
  • Sandwich theorem
  • Limit superior and limit inferior
  • Sub-sequence
  • Some theorems of sub-sequence
  • Cauchy's sequence or fundamental sequence
  • Some important theorems of Cauchy's sequence
  • Cauchy's general principle of convergence for sequence
  • Cauchy's first theorem on limits
  • Cauchy's second theorem on limits
  • Cesaro's theorem

Infinite Series

  • Introduction
  • Sequence of partial sums of series
  • Nature of an infinite series
  • Some important theorems
    • Cauchy's general principle of convergence Test of the convergence of geometric series
  • Comparative tests of the first type
  • Comparative tests of the second type
    • Ratio-comparison test; D' Alembert's ratio test; Raabe's test; de Morgan's and Bertrand's test; Logarithmic ratio test; Second logarithmic ratio test; Gauss's test
  • Some other useful tests
    • Cauchy's nth root test; Cauchy's condensation test
  • Alternating series
  • Absolute convergence
  • Conditionally convergence

Uniform Convergence

  • Introduction
  • Pointwise convergence of a sequence of functions
  • Uniform convergence
  • Series of functions
  • Cauchy's criterion for uniform convergence
  • Test for uniform convergence of a sequence and series of functions
  • Uniform convergence and continuity
  • Term by term integration
  • A sufficient condition for term by term integration of an infinite series
  • A sufficient condition for term by term differentiation of the series

Improper Integrals

  • Finite and infinite intervals
  • Bounded function
  • Improper integral
  • Types of improper integral
  • Convergence of improper integral of first kind
  • Convergence tests for the improper integral of first kind
  • Convergence of improper integral of second kind
  • Convergence tests for the improper integral of second kind
  • Convergence of improper integrals of third kind

Riemann Integration

  • Introduction
  • Partition of a closed interval
  • Norm of partition
  • Refinement of a partition
  • Supremum and infimum
  • Upper and lower Darboux sum
  • Theorems on Darboux sum
  • Upper and lower Riemann integral
  • Integral function
  • Riemann integral
  • Theorems of Riemann integral
    • Necessary and sufficient condition for a function to be R-integrable
  • Particular classes of Riemann integrable functions
  • Riemann integral as the limit of a sum
  • Properties of Riemann integral function
  • Integral function
  • Properties of integral function
  • Primitive
  • Mean value theorems of integral calculus
  • Fundamental theorem of integral
  • Techniques of integration

Fourier Series

  • Introduction
  • Perodic functions
  • Properties of definite integral
  • Some important definite integrals
  • Fourier series
  • Dirichlet's conditions for the expansion of a Fourier series
  • Even and odd functions
  • Fourier series for even and odd functions
  • Fourier's half range series
  • Other forms of Fourier series
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