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＜電子ブック＞
A Short Book on Long Sums : Infinite Series for Calculus Students / by Fernando Q. Gouvêa
(Readings in Mathematics. ISSN:29455847)

版	1st ed. 2023.
出版者	(Cham : Springer Nature Switzerland : Imprint: Springer)
出版年	2023
本文言語	英語
大きさ	XI, 145 p. 48 illus., 41 illus. in color : online resource
著者標目	*Gouvêa, Fernando Q author SpringerLink (Online service)
件　名	LCSH:Sequences (Mathematics) LCSH:Mathematical analysis LCSH:Functions of real variables FREE:Sequences, Series, Summability FREE:Analysis FREE:Real Functions
一般注記	- To the reader -- Getting close with lines -- Getting closer with polynomials -- Going all the way: Convergence -- Power series -- Distant mountains -- Appendix A: SageMath: A (very) short introduction -- Appendix B: Why I do it this way -- Bibliography 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 HTTP:URL=https://doi.org/10.1007/978-3-031-37557-6

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