Calculus of Real Series and Convergence for University Entrance Exams

Succeeding in competitive mathematics exams requires a rock-solid grasp of real analysis and series behavior. This comprehensive textbook-style course breaks down complex mathematical proofs and convergence tests into logical, readable explanations. You will transition from basic sequence definitions to solving advanced series problems with confidence and precision. Learn to identify convergence patterns, apply the correct testing methodologies, and avoid common algebraic traps. What you will learn: Understand foundational concepts of sequences, limits, and the completeness of real numbers; Apply core convergence tests including the Ratio, Root, Comparison, and Integral tests; Analyze alternating series using Leibniz's test and determine absolute versus conditional convergence; Master power series expansion, finding interval and radius of convergence; Practice structured problem-solving techniques tailored for university entrance examinations. The course begins with essential definitions of sequences and limits, establishing a strong theoretical foundation. From there, you will read through step-by-step mathematical proofs, exploring how to analyze diverse series types through clear, written examples. This course is designed for undergraduate mathematics students and aspirants preparing for competitive entrance exams, requiring only a basic background in high school calculus. Begin reading today to sharpen your mathematical reasoning and excel in your upcoming examinations.