Introduction to Calculus and Classical Analysis

This is the second edition of an undergraduate one-variable analysis text. Apart from correcting errors and rewriting several sections, material has been added, notably in Chapter 1 and Chapter 4. A noteworthy addition is a re- variable computation of the radius of convergence of the Bernoulli series using the root test (Chapter 5). What follows is the preface from the ? rst edition. For undergraduate students, the transition from calculus to analysis is often disorienting and mysterious. What happened to the beautiful calculus formulas? Wheredid -? andopensetscomefrom? Itisnotuntillaterthatone integrates these seemingly distinct points of view. When teaching “advanced calculus”, I always had a di? cult time answering these questions. Now, everymathematicianknowsthatanalysisarosenaturallyintheni- teenthcenturyoutofthecalculusoftheprevioustwocenturies. Believingthat it was possible to write a book re? ecting, explicitly, this organic growth, I set outtodoso. I chose several of the jewels of classical eighteenth and nineteenth century analysisandinsertedthemattheendofthebook, insertedtheaxiomsforreals at the beginning, and ? lled in the middle with (and only with) the material necessaryforclarityandlogical completeness. Intheprocess, everylittle piece of one-variable calculus assumed its proper place, and theory and application were interwoven throughout.