This text contains material for a two- or three-semester undergraduate course. The aim is to sketch the logical and mathematical underpinnings of the theory of series and one-variable calculus, develop that theory rigorously, and pursue some of its refinements and applications in the direction of measure theory, Fourier series, and differential equations.

A good working knowledge of calculus is assumed. Some familiarity with vector spaces and linear transformations is desirable but, for most topics, is not indispensable.

The unstarred sections are the core of the course. They are largely independent of the starred sections. The starred sections, on the other hand, contain some of the most interesting material.

Solving problems is an essential part of learning mathematics. Hints are given at the end for most of the exercises, but a hint should be consulted only after a real effort has been made to solve the problem.

I am grateful to various colleagues, students, friends, and family members for comments on, and corrections to, various versions of the notes that preceded this book. Walter Craig enlightened me about the difference between clarinets and oboes, and the consequences of that difference. Eric Belsley provided numerous corrections to the first version of the notes for Chapters 1–9. Other helpful comments and corrections are due to Stephen Miller, Diana Beals-Reid, and Katharine Beals. Any new or remaining mistakes are my responsibility.

I had the privilege of first encountering many of these topics in a course taught by Shizuo Kakutani, to whom this book is respectfully dedicated.