Taylor Polynomials and Infinite Series A chapter for a first-year calculus course By Benjamin Goldstein Table of Contents Preface...................................................................................................................................... iii Section 1 – Review of Sequences and Series ...............................................................................1 Section 2 – An Introduction to Taylor Polynomials ...................................................................18 Section 3 – A Systematic Approach to Taylor Polynomials .......................................................34 Section 4 – Lagrange Remainder...............................................................................................48 Section 5 – Another Look at Taylor Polynomials (Optional) .....................................................62 Section 6 – Power Series and the Ratio Test..............................................................................66 Section 7 – Positive-Term Series...............................................................................................81 Section 8 – Varying-Sign Series................................................................................................96 Section 9 – Conditional Convergence (Optional).....................................................................112 Section 10 – Taylor Series.......................................................................................................116 © 2013 Benjamin Goldstein Preface The subject of infinite series is delightful in its richness and beauty. From the "basic" concept that adding together infinitely many things can sometimes lead to a finite result, to the extremely powerful idea that functions can be represented by Taylor series–"infinomials," as one of my students once termed them–this corner of the calculus curriculum covers an assortment of topics that I have always found fascinating. This text is a complete, stand-alone chapter covering infinite sequences and series, Taylor polynomials, and power series that attempts to make these wonderful topics accessible and understandable for high school students in an introductory calculus course. It can be used to replace the analogous chapter or chapters in a standard calculus textbook, or it can be used as a secondary resource for students who would like an alternate narrative explaining how these topics are connected. There are countless calculus textbooks on the market, and it is the good fortune of calculus students and teachers that many of them are quite good. You almost have to go out of your way to pick a bad introductory calculus text. Why then am I adding to an already-glutted market my own chapter on Taylor polynomials and Taylor series? For my tastes, none of the available excellent texts organize or present these topics quite the way I would like. Not without reason, many calculus students (and not a small number of their teachers) find infinite series challenging. They deserve a text that highlights the big ideas and focuses their attention on the key themes and concepts. The organization of this chapter differs from that of standard texts. Rather than progress linearly from a starting point of infinite sequences, pass through infinite series and convergence tests, and conclude with power series generally and Taylor series specifically, my approach is almost the opposite. The reason for the standard approach is both simple and clear; it is the only presentation that makes sense if the goal is to develop the ideas with mathematical rigor. The danger, however, is that by the time the students have slogged through a dozen different convergence tests to arrive at Taylor series they may have lost sight of what is important: that functions can be approximated by polynomials and represented exactly by "infinitely long polynomials." So I start with the important stuff first, at least as much of it as I can get away with. After an obligatory and light-weight section reviewing sequences and series topics from precalculus, I proceed directly to Taylor polynomials. Taylor polynomials are an extension of linearization functions, and they are a concrete way to frame the topics that are to come in the rest of the chapter. The first half of the chapter explores Taylor polynomials—how to build them, under what conditions they provide good estimates, error approximation—always with an eye to the Taylor series that are coming down the road. The second half of the chapter extends naturally from polynomials to power series, with convergence tests entering the scene on an as-needed basis. The hope is that this organizational structure will keep students thinking about the big picture. Along the way, the book is short on proofs and long on ideas. There is no attempt to assemble a rigorous development of the theory; there are plenty of excellent textbooks on the market that already do that. Furthermore, if we are to be honest with each other, students who want to study these topics with complete mathematical precision will need to take an additional course in advanced calculus anyway. A good text is a resource, both for the students and the teacher. For the students, I have given this chapter a conversational tone which I hope makes it readable. Those of us who have been teaching a while know that students don't always read their textbooks as much as we'd like. (They certainly don't read prefaces, at any rate. If you are a student reading this preface, then you are awesome.) Even so, the majority of the effort of this chapter was directed toward crafting an exposition that makes sense of the big ideas and how they fit together. I hope that your students will read it and find it helpful. For teachers, a good text provides a bank of problems. In addition to standard problem types, I have tried to include problems that get at the bigger concepts, that challenge students to think in accordance with the rule of four (working with mathematical concepts verbally, numerically, graphically, and analytically), and that provide good preparation for that big exam that occurs in early May. The final role a math text should fill if it is to be a good resource is to maintain a sense of the development of the subject, presenting essential iii theorems in a logical order and supporting them with rigorous proofs. This chapter doesn't do that. Fortunately, students have their primary text, the one they have been using for the rest of their calculus course. While the sequencing of the topics in this chapter is very different from that of a typical text, hopefully the student interested in hunting down a formal proof for a particular theorem or convergence test will be able to find one in his or her other book. Using this chapter ought to be fairly straightforward. As I have said, the focus of the writing was on making it user-friendly for students. The overall tone is less formal, but definitions and theorems are given precise, correct wording and are boxed for emphasis. There are numerous example problems throughout the sections. There are also practice problems. Practices are just like examples except that their solutions are delayed until the end of the section. Practices often follow examples, and the hope is that students will work the practice problems as they read to ensure that they are picking up the essential ideas and skills. The end of the solution to an example problem (or to the proof of a theorem) is signaled by a ◊. Parts of the text have been labeled as "optional." The optional material is not covered on the AP test, nor is it addressed in the problem sets at the ends of the sections. This material can be omitted without sacrificing the coherence of the story being told by the rest of the chapter. There are several people whose contributions to this chapter I would like to acknowledge. First to deserve thanks is a man who almost certainly does not remember me as well as I remember him. Robert Barefoot was the first person (though not the last!) to suggest to me that this part of the calculus curriculum should be taught with Taylor polynomials appearing first, leaving the library of convergence tests for later. Much of the organization of this chapter is influenced by the ideas he presented at a workshop I attended in 2006. As for the actual creation of this text, many people helped me by reading drafts, making comments, and discussing the presentation of topics in this chapter. I am very grateful to Doug Kühlmann, Melinda Certain, Phil Certain, Beth Gallis, Elisse Ghitelman, Carl LaCombe, Barbara De Roes, James King, Sharon Roberts, Stephanie Taylor, and Scott Barcus for their thoughtful comments, feedback, error corrections, and general discussion on the composition and use of the chapter. Scott was also the first to field test the chapter, sharing it with his students before I even had a chance to share it with mine. I am certainly in debt to my wife Laura who proof-read every word of the exposition and even worked several of the problems. Finally, I would be remiss if I did not take the time to thank Mary Lappan, Steve Viktora, and Garth Warner. Mary and Steve were my first calculus teachers, introducing me to a subject that I continue to find more beautiful and amazing with every year that I teach it myself. Prof. Warner was the real analysis professor who showed me how calculus can be made rigorous; much of my current understanding of series is thanks to him and his course. The subject of infinite series can be counter-intuitive and even