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.