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Book Summary
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This book has three chapters. In the first chapter, the characteristic difference between the set of real numbers and rational numbers has been discussed with regard to completeness property. The concept of neighbourhood open and closed sets, limit points, have been incorporated leading to Bolzano-Weistrass's theorem. Final part of this chapter discusses continuous functions and their properties and uniform continuity over a domain. In the second chapter, sequences and series have been studied ascertaining their convergence and divergence using various tests, like basic comparison tests, limit comparison test, D'Alembert's ratio test, Cauchy's nth root test, Raabe's test, Cauchy's integral test, etc. Leibnitz's test for alternating series, absolute and conditional convergence, subsequences, limit superior and limit inferior and properties of monotone sequences have been discussed in detail. The third chapter deals with Riemann integration giving emphasis to the proofs of the necessary and sufficient conditions of Riemann integrability of bounded functions. The integrability of continuous functions and monotonic function have been proved in order to enable students to identify functions which are integrable.
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Book Content
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- Properties of Real Numbers, Continuous Functions and Uniformly Continuous Functions
- Sequence and Series
- Riemann Integral
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