Sequence and Series of Real Numbers for Competitive Mathematics

Mastering the behavior of sequences and series is a fundamental milestone for anyone pursuing advanced mathematics or preparing for rigorous competitive exams. This text-only course provides a comprehensive, step-by-step guide to the mathematical theory of real sequences and series. You will transition from basic definitions to advanced convergence tests, developing the analytical skills needed to solve complex calculus and real analysis problems with absolute confidence. What you'll learn: - Understand the fundamental properties of real numbers, bounds, and the completeness axiom - Analyze the convergence, divergence, and oscillation of real sequences using formal definitions - Apply key convergence tests including the Ratio, Root, Comparison, and Integral tests - Determine limit superior, limit inferior, and the behavior of monotonic sequences - Evaluate alternating series using Leibniz's test and understand absolute versus conditional convergence - Practice solving rigorous mathematical proofs and exam-style problems through detailed written walkthroughs We begin with foundational concepts of real numbers and limits before exploring advanced convergence behavior and series expansions. Each module features clear explanations, step-by-step proofs, and practical exercises designed to reinforce your analytical thinking. Designed for undergraduate mathematics students, calculus learners, and candidates preparing for competitive exams, this course requires only a basic background in algebra. Start reading today to build a flawless foundation in real analysis.