Real Analysis Volume 2 The Riemann Integral and the Convergence of Functions (Bachelor's Degree in Mathematics)

Real Analysis: Volume 2 - Integration and Convergence is the ideal continuation for students, instructors, and researchers seeking a rigorous treatment of mathematical analysis. This book progressively develops the foundations of the Riemann Integral and the convergence of sequences and series of functions, providing a solid theoretical basis supported by formal proofs and a wide collection of solved and proposed exercises.
What will you find in this book?
Riemann Integral and Its Properties
A rigorous formalization of the integral as the limit of Riemann sums. Fundamental properties such as linearity, monotonicity, and additivity are studied, along with applications in calculus and mathematical analysis.
Fundamental Theorem of Calculus
An exploration of the relationship between differentiation and integration. Applications in physics and mathematics are presented, establishing the essential link between both operations.
Improper Integrals and Convergence Criteria
A detailed analysis of improper integrals, their classification, and conditions for convergence. Key examples are included in applied mathematics and probability problems.
Pointwise and Uniform Convergence
Definition and comparison between pointwise and uniform convergence, with fundamental theorems and detailed examples. The impact of these notions on the continuity, differentiability, and integrability of limit functions is studied.
Power Series and Function Approximation
Development of the theory of power series and its importance in the representation of functions. Convergence criteria such as Abel's and Dirichlet's tests are included, with applications in differential equations and complex analysis.
Solved and Proposed Exercises
Each chapter presents a broad selection of problems carefully designed to reinforce theoretical understanding and enhance mathematical intuition.
Highlights:
✔ Rigorous and detailed explanations: Ideal for those seeking structured and in-depth learning.
✔ Formal examples and proofs: Each concept is presented with well-founded theorems and applications in mathematical contexts.
✔ Didactic and progressive approach: From fundamental definitions to advanced results in integration and convergence.
Who is this book for?
Undergraduate and graduate students in mathematics and exact sciences seeking a formal reference in real analysis.
Instructors and self-learners who need structured material with rigorous theory and challenging exercises.
Researchers and professionals looking for a solid text on integration, convergence, and functional analysis.

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