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This is a widely accessible introductory treatment of infinite series of real numbers, bringing the reader from basic definitions and tests to advanced results. An up-to-date presentation is given, making infinite series accessible, interesting, and useful to a wide audience, including students, teachers, and researchers. Included are elementary and advanced tests for convergence or divergence, the harmonic series, the alternating harmonic series, and closely related results. One chapter offers 107 concise, crisp, surprising results about infinite series. Another gives problems on infinite series, and solutions, which have appeared on the annual William Lowell Putnam Mathematical Competition. The lighter side of infinite series is treated in the concluding chapter where three puzzles, eighteen visuals, and several fallacious proofs are made available. Three appendices provide a listing of true or false statements, answers to why the harmonic series is so named, and an extensive list of published works on infinite series.
Careful presentation of fundamentals of the theory by one of the finest modern expositors of higher mathematics. Covers functions of real and complex variables, arbitrary and null sequences, convergence and divergence, Cauchy's limit theorem, more.
This textbook covers the majority of traditional topics of infinite sequences and series, starting from the very beginning – the definition and elementary properties of sequences of numbers, and ending with advanced results of uniform convergence and power series. The text is aimed at university students specializing in mathematics and natural sciences, and at all the readers interested in infinite sequences and series. It is designed for the reader who has a good working knowledge of calculus. No additional prior knowledge is required. The text is divided into five chapters, which can be grouped into two parts: the first two chapters are concerned with the sequences and series of numbers, while the remaining three chapters are devoted to the sequences and series of functions, including the power series. Within each major topic, the exposition is inductive and starts with rather simple definitions and/or examples, becoming more compressed and sophisticated as the course progresses. Each key notion and result is illustrated with examples explained in detail. Some more complicated topics and results are marked as complements and can be omitted on a first reading. The text includes a large number of problems and exercises, making it suitable for both classroom use and self-study. Many standard exercises are included in each section to develop basic techniques and test the understanding of key concepts. Other problems are more theoretically oriented and illustrate more intricate points of the theory, or provide counterexamples to false propositions which seem to be natural at first glance. Solutions to additional problems proposed at the end of each chapter are provided as an electronic supplement to this book.
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Trans from the 2nd German ed , pub 1923.
Text for advanced undergraduate and graduate students examines Taylor series, Fourier series, uniform convergence, power series, and real analytic functions. Appendix covers set and sequence operations and continuous functions. 1962 edition.
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An informal and practically focused introduction for undergraduate students exploring infinite series and sequences in engineering and the physical sciences. With a focus on practical applications in real world situations, it helps students to conceptualize the theory with real-world examples and to build their skill set.
A telekinetic assassin's quest to avenge his parents' murders triggers an all-out war between an eccentric cast of killers, from the New York Times bestselling author of INFINITE and THE DARK. Jonas kills people...with his mind. He reaches out, focuses for a moment, and feels a snap of pain between his eyes. Then his target falls over dead, a hole in the center of their brains. The cause of death is unknown to all aside from Jonas, and his lone companion: Bubbles, a sarcastic AI with aspirations to be more human, or a serial killer. One of the two. Maybe both. Jonas calls his ability... MIND BULLET Sensing Jonas is lonely and depressed, Bubbles arranges for Madee, a Thai delivery woman by day, and a thief by night, to enter his life. For a moment: sparks. But chaos follows as the people Madee last robbed catch up to her. They also happen to be the Shrieking Ninjas. After surviving the assault, Jonas and Madee find themselves on the run, and a 10 million dollar hit called on Jonas. Killers from all over the globe pursue the pair as they run for their lives and headlong into Jonas's mysterious past, uncovering the surreal origins and extent of his abilities. They discover a dark and twisted history that threatens to unravel his mind and leave him susceptible to control. Working with Bubbles, Madee, and a growing list of new allies, Jonas battles his way through waves of assassins, some with abilities like his, all to protect the world from a recently discovered, untraceable weapon of mass destruction-himself. MIND BULLET is a humor-fueled, edge-of-your-seat thriller that takes the reader to the brink of sanity and then drops them into the abyss. Science fiction and mythology collide in this story about the power of family, and the role parents play in our lives-both good and evil.