Foundational Real Analysis: Sequences and Infinite Series

Sequences and series are the building blocks of calculus and advanced mathematical analysis, yet their rigorous definitions can often be confusing. This course demystifies the concepts behind limits, convergence, and infinite sums. By the end of this comprehensive text-based course, you will possess a deep, formal understanding of sequences and series, allowing you to confidently tackle advanced problems in real analysis and apply precise mathematical reasoning to determine convergence and divergence. What you'll learn: * Understand the formal Epsilon-N definition of sequence limits and convergence. * Practice applying the Monotone Convergence Theorem and Cauchy criterion for sequences. * Master the foundational tests for series convergence, including the Ratio, Root, and Comparison Tests. * Analyze the differences between absolute and conditional convergence for alternating series. * Build, manipulate, and apply power series, including Taylor and Maclaurin expansions. * Apply rigorous proof-writing techniques essential for advanced mathematical study. The course begins with foundational concepts and rigorous definitions before progressing to practical tests for convergence and detailed analysis of infinite series, culminating in an exploration of power series expansions. Written explanations and detailed solved examples guide you through every concept. This course is designed for absolute beginners in advanced mathematics, students transitioning from calculus to real analysis, or anyone needing a rigorous refresher on sequences and series. No prior knowledge of formal proof writing is required. Start reading today and build a robust foundation in mathematical analysis.