Real Analysis via Sequences and Series
Book:Real Analysis via Sequences and Series
Authors: Charles H. C. Little, Kee L. Teo, Bruce van Brunt
Publication date: 28 May 2015 (eBook)
Publisher: Springer, New York
Series:Undergraduate Texts in Mathematics
Summary
Real Analysis via Sequences and Series is an undergraduate introduction to rigorous real analysis that takes a somewhat different route from many traditional textbooks. Instead of beginning primarily with limits of functions and the $\varepsilon$–$\delta$ formalism, the authors make sequences and infinite series the central organizing ideas. Once convergence of sequences and series has been developed carefully, the same viewpoint is used to construct the standard theory of limits, continuity, differentiation, Riemann integration, Taylor series, fixed points, and sequences of functions.
The progression is therefore particularly natural for students moving from computational calculus toward proof-based mathematics. The book contains motivated definitions, rigorous proofs, many worked examples and counterexamples, and exercises after most sections. Its treatment of infinite series is especially substantial, covering numerous convergence tests as well as absolute and conditional convergence. It also goes beyond the minimum syllabus with attractive applications and classical results such as Wallis's formula, Stirling's formula, proofs of the irrationality of $e$ and $\pi$, and Newton's method interpreted as a fixed-point iteration.
The later chapters bring the reader into recognizably modern real analysis: the Riemann integral, Taylor polynomials and series, fixed-point problems, and sequences of functions, including the ideas needed to understand uniform convergence. Reviewers describe it as a well-written text suitable for a first university course in mathematical analysis, particularly for upper-level undergraduates learning mathematical rigor for the first time.
Key takeaways
BOOK
Book:Real Analysis via Sequences and Series
Authors: Charles H. C. Little, Kee L. Teo, Bruce van Brunt
Publication date: 28 May 2015 (eBook)
Publisher: Springer, New York
Series:Undergraduate Texts in Mathematics
Summary
Real Analysis via Sequences and Series is an undergraduate introduction to rigorous real analysis that takes a somewhat different route from many traditional textbooks. Instead of beginning primarily with limits of functions and the $\varepsilon$–$\delta$ formalism, the authors make sequences and infinite series the central organizing ideas. Once convergence of sequences and series has been developed carefully, the same viewpoint is used to construct the standard theory of limits, continuity, differentiation, Riemann integration, Taylor series, fixed points, and sequences of functions.
The progression is therefore particularly natural for students moving from computational calculus toward proof-based mathematics. The book contains motivated definitions, rigorous proofs, many worked examples and counterexamples, and exercises after most sections. Its treatment of infinite series is especially substantial, covering numerous convergence tests as well as absolute and conditional convergence. It also goes beyond the minimum syllabus with attractive applications and classical results such as Wallis's formula, Stirling's formula, proofs of the irrationality of $e$ and $\pi$, and Newton's method interpreted as a fixed-point iteration.
The later chapters bring the reader into recognizably modern real analysis: the Riemann integral, Taylor polynomials and series, fixed-point problems, and sequences of functions, including the ideas needed to understand uniform convergence. Reviewers describe it as a well-written text suitable for a first university course in mathematical analysis, particularly for upper-level undergraduates learning mathematical rigor for the first time.
Key takeaways
- Main subject: Real Analysis / Mathematical Analysis.
- Central idea: Develop analysis from sequences and series, rather than treating them as secondary topics.
- Core progression:
Sequences→Series→Limits→Continuity→Differentiability→Integration.$\text{Sequences}\rightarrow\text{Series}\rightarrow\text{Limits}\rightarrow\text{Continuity}\rightarrow\text{Differentiability}\rightarrow\text{Integration}$.
- The book contains particularly extensive material on convergence of infinite series.
- Later topics include Taylor series, fixed-point theory, Newton's method, and sequences of functions.
- Interesting classical results—including Stirling's and Wallis's formulas and the irrationality of $e$ and $\pi$—give the book more mathematical character than a purely standard calculus-to-analysis text.
- Level: roughly advanced undergraduate; appropriate as a student's first rigorous real-analysis course.
- Mathematical value: particularly good for bridging the gap between elementary calculus and more abstract courses such as measure theory, functional analysis, differential equations, and advanced analysis.
BOOK
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