Foundations of Mathematical Analysis (Google eBook)
Mathematical analysis is fundamental to the undergraduate curriculum not only because it is the stepping stone for the study of advanced analysis, but also because of its applications to other branches of mathematics, physics, and engineering at both the undergraduate and graduate levels. This self-contained textbook consists of eleven chapters, which are further divided into sections and subsections. Each section includes a careful selection of special topics covered that will serve to illustrate the scope and power of various methods in real analysis. The exposition is developed with thorough explanations, motivating examples, and illustrations conveying geometric intuition in a pleasant and informal style to help readers grasp difficult concepts. Key features include: * Questions and Exercises are provided at the end of each section, covering a broad spectrum of content with various levels of difficulty; * Some of the exercises are routine in nature while others are interesting, instructive, and challenging with hints provided for selected exercises; * Covers a broad spectrum of content with a range of difficulty that will enable students to learn techniques and standard analysis tools; * Introduces convergence, continuity, differentiability, the Riemann integral, power series, uniform convergence of sequences and series of functions, among other topics; * Examines various important applications throughout the book; * Figures throughout the book to demonstrate ideas and concepts are drawn using Mathematica. Foundations of Mathematical Analysis is intended for undergraduate students and beginning graduate students interested in a fundamental introduction to the subject. It may be used in the classroom or as a self-study guide without any required prerequisites.
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2 Sequences Convergence and Divergence
3 Limits Continuity and Differentiability
4 Applications of Differentiability
5 Series Convergence and Divergence
6 Definite and Indefinite Integrals
7 Improper Integrals and Applications of Riemann Integrals
8 Power Series
ak+1 akxk alternating series test an+1 bounded variation comparison test Consider continuous function convergent series converges absolutely converges uniformly Corollary cosx curve DarbouxIStieltjes decreasing deﬁned deﬁnition diverges dx converges endpoints Example Exercises exists f and g f is continuous f is differentiable f is integrable Figure ﬁnd ﬁnite ﬁrst ﬁxed Fourier series function f function of bounded given series graph hence implies improper integral increasing inequality inﬁnite interval Lemma Let f limit mean value theorem monotone nIoo Note obtain one-to-one partial sums partition P I power series Proof prove radius of convergence rational numbers real numbers Riemann integrable Riemann sum root test sequence of partial sequence of real series converges Show that f sinx squeeze rule summable Suppose that f uniform convergence uniformly continuous zero