A Concrete Introduction to Real Analysis
Most volumes in analysis plunge students into a challenging new mathematical environment, replete with axioms, powerful abstractions, and an overriding emphasis on formal proofs. This can lead even students with a solid mathematical aptitude to often feel bewildered and discouraged by the theoretical treatment. Avoiding unnecessary abstractions to provide an accessible presentation of the material, A Concrete Introduction to Real Analysis supplies the crucial transition from a calculations-focused treatment of mathematics to a proof-centered approach.
Drawing from the history of mathematics and practical applications, this volume uses problems emerging from calculus to introduce themes of estimation, approximation, and convergence. The book covers discrete calculus, selected area computations, Taylor's theorem, infinite sequences and series, limits, continuity and differentiability of functions, the Riemann integral, and much more. It contains a large collection of examples and exercises, ranging from simple problems that allow students to check their understanding of the concepts to challenging problems that develop new material.
Providing a solid foundation in analysis, A Concrete Introduction to Real Analysis demonstrates that the mathematical treatments described in the text will be valuable both for students planning to study more analysis and for those who are less inclined to take another analysis class.
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analysis approximation argument axioms ck converges coefficients computation consider continued fraction continuous at x0 converges absolutely convex deﬁned deﬁnition derivative domain endpoint Riemann sums equation estimate example f is continuous Figure ﬁrst formula function f Fundamental Theorem geometric series gives graph implies improper integrals induction infinite series integer interval a,b least upper bound Lemma lim x→x0 limit limk limn log(x lower sums mathematics Mean Value Theorem midpoint modus ponens Nested Interval Nested Interval Principle nonnegative open interval partial sums partition positive integer power series predicate problem Proof propositional logic rational numbers real numbers rectangles result Riemann integrable right endpoint Riemann satisfying sequence of partial sequence xk series converges Show statement subintervals Suppose that f Taylor polynomial Taylor series Theorem of Calculus true truth table truth value variables