{"title":"An Analysis of Sequence and Series","authors":"Yogendra Prasad Shah","doi":"10.3126/hj.v15i1.63974","DOIUrl":null,"url":null,"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.","PeriodicalId":47458,"journal":{"name":"Historical Journal","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Historical Journal","FirstCategoryId":"98","ListUrlMain":"https://doi.org/10.3126/hj.v15i1.63974","RegionNum":1,"RegionCategory":"历史学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"HISTORY","Score":null,"Total":0}
引用次数: 0
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.
本文全面探讨了序列和数列这一应用广泛的数学基本概念。文章从介绍序列开始，包括算术序列和几何序列，进而讨论有限序列和 n 元组序列。文章探讨了算术数列和几何数列及其和的公式。文章深入探讨了序列和数列的性质，讨论了收敛、发散和绝对收敛。文章定义了有界数列和单调数列等各种类型的数列，并介绍了作为数列极限存在性判据的考希判据。此外，还涉及有关序列的基本事实，包括绝对收敛性和作为序列之和的数 e。此外，还介绍了函数序列的概念。文章最后强调了序列和数列在高等数学、艺术、科学、技术和金融等领域的重要意义，展示了它们在不同学科的决策、金融分析和风险评估中的关键作用。
期刊介绍：
The Historical Journal continues to publish papers on all aspects of British, European, and world history since the fifteenth century. The best contemporary scholarship is represented. Contributions come from all parts of the world. The journal aims to publish some thirty-five articles and communications each year and to review recent historical literature, mainly in the form of historiographical reviews and review articles. The journal provides a forum for younger scholars making a distinguished debut as well as publishing the work of historians of established reputation.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
