An Analysis of Sequence and Series

Abstract

This article provides a comprehensive exploration of sequences and series, fundamental concepts in mathematics with wide-ranging applications. Starting with an introduction to sequences, including arithmetic and geometric sequences, the discussion extends to finite sequences and n-tuple sequences. Arithmetic and geometric series are explored, with formulas for their sums. The article delves into the properties of sequences and series, discussing convergence, divergence, and absolute convergence. Various types of sequences, such as bounded and monotonic sequences, are defined, and the Cauchy Criterion is introduced as a criterion for the existence of the limit of a sequence. Elementary facts about series, including absolute convergence and the number e as the sum of a series, are covered. Additionally, the concept of sequences of functions is introduced. The article concludes by emphasizing the significance of sequences and series in fields like higher mathematics, art, science, technology, and finance, showcasing their crucial role in decision-making, financial analysis, and risk assessment across diverse disciplines.