Recent Advancements in Sequences and Series of Real Numbers: Convergence, Fractals, Chaos Theory, and Applications in Dynamical Systems

Abstract

This study explores the fundamental concepts of sequences and series of real numbers, focusing on their mathematical properties, convergence criteria, and applications. We analyze the behavior of sequences, defining their limits and types, and delve into the theory of series, particularly infinite series, with a focus on convergence tests such as the ratio test, root test, and comparison test. It examines the mathematical foundations and significance of these concepts in real analysis. Additionally, the study investigates the role of fractal sequences in chaos theory, emphasizing their application in understanding complex dynamical systems.