Real Analysis via Sequences and Series

Description

This text gives a rigorous treatment of the foundations of calculus. In contrast to more traditional approaches, infinite sequences and series are placed at the forefront. The approach taken has not only the merit of simplicity, but students are well placed to understand and appreciate more sophisticated concepts in advanced mathematics. The authors mitigate potential difficulties in mastering the material by motivating definitions, results and proofs. Simple examples are provided to illustrate new material and exercises are included at the end of most sections. Noteworthy topics include: an extensive discussion of convergence tests for infinite series, Wallis’s formula and Stirling’s formula, proofs of the irrationality of π and e and a treatment of Newton’s method as a special instance of finding fixed points of iterated functions. Front Matter....Pages i-xi Introduction....Pages 1-32 Sequences....Pages 33-108 Series....Pages 109-189 Limits of Functions....Pages 191-214 Continuity....Pages 215-241 Differentiability....Pages 243-332 The Riemann Integral....Pages 333-398 Taylor Polynomials and Taylor Series....Pages 399-421 The Fixed-Point Problem....Pages 423-436 Sequences of Functions....Pages 437-469 Back Matter....Pages 471-476