Worked examples (2) - Boundedness of Sequences | Convergence of Infinite Sequences and Series - Advanced Calculus (Undergraduate Advanced)

Do you want to learn how to work with infinite sums of numbers and functions and their properties, operations, and applications? Do you want to understand the concepts of progressions, sequences, series, and power series, and how they relate to the approximation, convergence, and divergence of functions? Do you want to master the skills of finding and applying various tests and criteria of convergence and divergence to different types of series, such as arithmetic, geometric, harmonic, p-series, and alternating series? If you answered yes to any of these questions, then this course is for you! This course reviews the fundamental concepts of finite progressions and provides a thorough treatise of the convergence of infinite real sequences and series. You will learn how to: - Define and classify progressions and their properties, such as common difference, common ratio, and sum to infinity - Define and classify sequences and their properties, such as terms, general term, and boundedness - Find the convergence or divergence of a sequence using the formal definition and various theorems and examples - Define and classify series and their properties, such as partial sums, absolute and conditional convergence, and uniform convergence - Find the convergence or divergence of a series of real numbers using various methods and techniques, such as direct comparison, integral test, p-series test, ratio test, Raabe's test, and alternating series test - Define and classify sequences and series of functions and their properties, such as pointwise and uniform convergence, and term-by-term differentiation and integration - Find the convergence or divergence of a sequence or series of functions using various methods and techniques, such as Weierstrass M-test, Abel's test, and Dirichlet's test - Define and classify power series and their properties, such as radius and interval of convergence, and term-by-term differentiation and integration - Find the convergence or divergence of a power series using various methods and techniques, such as ratio test, root test, and Cauchy-Hadamard theorem This course is suitable for anyone who wants to learn or review the advanced topics of calculus and its applications. It is especially useful for students and professionals in analysis, differential equations, optimization, physics, engineering, and other related fields. By the end of this course, you will have a solid foundation of the theory and practice of calculus and its operations. You will also be able to apply the knowledge and skills you learned to real-world problems and challenges that involve infinite sums of numbers and functions and their approximation, convergence, and divergence. Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.