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A Short Book on Long Sums

Infinite Series for Calculus Students

  • Textbook
  • © 2023


  • Discusses infinite and power series through an informal, captivating narrative
  • Offers problems specially crafted to foster understanding instead of memorization
  • Serves as a supplementary reading for courses such as Real Analysis and Methods of Applied Mathematics
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Part of the book series: Undergraduate Texts in Mathematics (UTM)

Part of the book sub series: Readings in Mathematics (READINMATH)

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About this book

This concise textbook introduces calculus students to power series through an informal and captivating narrative that avoids formal proofs but emphasizes understanding the fundamental ideas. Power series—and infinite series in general—are  a fundamental tool of pure and applied mathematics. The problems focus on ideas, applications, and creative thinking instead of being repetitive and procedural. 

Calculus is about functions, so the book turns on two fundamental ideas: using polynomials to approximate a function and representing a function in terms of simpler functions. The derivative is reinterpreted in terms of linear approximations, which then leads to Taylor polynomials and the question of convergence. Enough of the theory of convergence is developed to allow a more complete understanding of power series and their applications. A final chapter looks at the distant horizon and discusses other kinds of series representations. SageMath, a free open-source mathematics software system, is used throughout to do computations, provide examples, and create many graphs. While most problems do not require SageMath, students are encouraged to use it where appropriate. An instructor’s guide with solutions to all the problems is available. 

The book is intended as a supplementary textbook for calculus courses; lecturers and instructors will find innovative and engaging ways to teach this topic. The informal and conversational tone make the book useful to any student seeking to understand this essential aspect of analysis.


Table of contents (5 chapters)

Authors and Affiliations

  • Department of Mathematics, Colby College, Waterville, USA

    Fernando Q. Gouvêa

About the author

Fernando Q. Gouvêa is a number theorist and historian of mathematics. In number theory, he has been interested in the connection between p-adic modular forms and deformations of Galois representations. As a historian, his main focus is on the early history of algebraic number theory, but he has also written on other historical topics. He enjoys books and has written many book reviews for a wide range of publications. For many years, he served as an editor of MAA Focus and MAA Reviews. His current project is a forthcoming book on the history of p-adic numbers and p-adic analysis in the first decades of the twentieth century. Gouvêa’s other books include Arithmetic of p-adic Modular Forms, A Guide to Groups, Rings, and Fields, and (with William P. Berlinghoff) Math through the Ages: A Gentle History for Teachers and Others.

Bibliographic Information

  • Book Title: A Short Book on Long Sums

  • Book Subtitle: Infinite Series for Calculus Students

  • Authors: Fernando Q. Gouvêa

  • Series Title: Undergraduate Texts in Mathematics

  • DOI:

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

  • Hardcover ISBN: 978-3-031-37556-9Published: 08 December 2023

  • Softcover ISBN: 978-3-031-37559-0Due: 08 January 2024

  • eBook ISBN: 978-3-031-37557-6Published: 07 December 2023

  • Series ISSN: 0172-6056

  • Series E-ISSN: 2197-5604

  • Edition Number: 1

  • Number of Pages: XI, 145

  • Number of Illustrations: 7 b/w illustrations, 41 illustrations in colour

  • Topics: Sequences, Series, Summability, Analysis, Real Functions

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