http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
Resources for the study of real analysis
"This book is a collection of materials gathered by the author while teaching real analysis over a period of years. It is intended for use as a supplement to a traditional analysis textbook, or to provide material for seminars or independent study in analysis and its historical development. The book includes historical and biographical information, a wide range of problem types, selected readings on a variety of topics, and many references for additional study. Since all these materials are collected into a single book, teachers and students can easily choose items most suitable for their purpose. Teachers may use the book as a supplement to their courses, while students may read much of the book on their own. No other book has been written specifically as a supplement for a real analysis course."--Publisher
Review of Calculus -- Analysis Problems -- Essays.
I. Review of calculus. An outline of a traditional course in calculus -- Calculus review problems -- II. Analysis problems. Basic problems. Properties of even and odd functions -- Use of the cancellation principle -- Finding Maclaurin series representations -- Evaluating ... by Cavalieri sums -- Supplementary problems. The pervasive nature of sequences -- Counterexamples -- Some examples of unusual functions -- Identifying interior, exterior, boundary, and limit points -- Exploring uniform continuity -- Comparing the ratio test, the root test, and Raabe's test -- Various proofs ... -- Finding a value for ... -- Rearrangements of conditionally convergent series -- Finding approximations for [pi]-- A function with derivatives of all orders, but no Maclaurin series -- Evaluating ... by Fermat's method -- Two proofs that ... -- Evaluating ... by Bernoulli's method -- Some interesting sequences of functions -- Comparing uniform continuity and uniform convergence -- Questions for sequences of functions and for series of functions -- Enrichment problems. Generalizing the Fibonacci sequence -- Euler's proof the e is irrational -- Subseries of the harmonic series -- Finding functions whose Maclaurin series is ... -- A pascal-like triangle -- Finding areas without antiderivatives -- Questions from the Euler-Stirling correspondence -- The Gamma function -- Bernoulli numbers -- A look at Fourier series -- Summability of divergent series -- Lebesque measure -- Cantor's middle third set -- III. Essays. History and biography. The time of Archimedes and other giants (pre-1660) -- The time of Newton and Leibniz (1660-1690) -- The time of Euler and the Bernoullis (1690-1790) -- The time of Cauchy, Fourier, and Lagrange (1790-1850) -- The time of Riemann and Weierstrass (1850-1900) -- The time of Lebesgue and Hardy (1900-1940) -- New looks at calculus content. Obtaining the derivative formulas -- Tests for convergence of series -- General topics for analysis. Proof techniques in analysis -- Sets and topology -- IV. Selected readings. John Bernoulli and the Marquis de l'Hôpital-- The Euler-Stirling correspondence -- Developing rigor in calculus -- The contribution of Lebesgue -- The Bernoulli numbers and some wonderful discoveries of Euler.
Includes bibliographical references and index.