Foundations of Real Analysis: Sequences and Series

Transitioning from computational calculus to rigorous mathematical analysis can feel daunting without a structured approach. This text-based course demystifies the foundational concepts of real analysis, focusing on the behavior of sequences and series. You will develop the mathematical maturity required to read, understand, and write rigorous proofs, transforming how you approach advanced mathematical problems. What you'll learn: - Understand the algebraic and topological properties of the real number system. - Analyze the convergence of sequences using formal definitions and limit laws. - Apply key convergence tests for infinite series, including the ratio, root, and comparison tests. - Explore modern applications of convergence in approximation algorithms and data science. - Practice constructing logical, rigorous mathematical proofs from first principles. You will start with the fundamental properties of real numbers before progressing systematically through limits, sequences, and various series convergence tests, supported by clear written explanations and exercises. This course is designed for undergraduate students, exam candidates, and self-learners seeking a solid foundation in mathematical analysis, with no prior proof-writing experience required. Start reading today to build your mathematical rigor and master real analysis.