数项级数求和方法的探究

数项级数求和方法的探究

摘要

级数，重要的数学工具。一方面，它对于数学和其他学科及技术研究与发展方面都起到了非常重要的作用并发挥了其重要影响。另一方面，级数还和我们的日常生活息息相关，我们要合理的掌握利用级数，也要去发掘更为广泛的应用领域，为日后的研究打下坚实的基础。

目前在国内并没有系统全面的研究级数求和的方法，级数求和首先要考虑级数的收敛性，并且有着比较繁多的方法和很强的技巧性。本文在借鉴国内外大量资料的基础上，选取了一些常用的数项级数求和的方法，如等差求和，等比求和，错位相减求和，原级数化为函数项级数、积分函数求和等，并且每种方法都选取了典型题目加以分析，尽量使理论与应用相结合，化繁为简。本文当中也特别介绍了如裂项法求和，夹逼法求和，幂级数求和等方法，并举出例子，在实例中说明方法，用实例体会这些方法在求和时的应用。

本文对数项级数的有关概念，收敛的定义做出了简要的说明。级数的敛散性是决定级数求和的先决条件，但是本文的重点在于讨论级数求和的方法，所以对级数敛散性的讨论略过不谈，并且本文中所提到的有关级数都是收敛的。

关键词：级数收敛数项级数求和裂项法求和幂级数求和

Abstract

Series, the important mathematical tools. On the one hand, it is for the mathematics and other science and technology research and development has played a very important role and exert important influence. On the other hand, the series also is closely linked with our daily life, we should reasonably grasp the use of series, also want to explore more widely used in the field, a solid foundation for the future research.

At present in domestic and no systematic research methods series comprehensive summation, summation of series should first consider the convergence of series, and have a comparison method and skills are very strong.In this paper, on the basis of numerous data home and abroad, some selected numerical series common summation method, such as the arithmetic sum, geometric sum, dislocation subtraction sum, positive numbers into the function series, integral function and so on, and each method selects a typical topic analysis as far as possible, so that the combination of theory and application, simplified.This paper also particularly introduces such as crack a summation, clamping force summation, and the method of power series, and examples, explain the method in the example, with the example of the application of these methods in the sum of.

In this paper, the related concepts of series, convergence definition and theorem gives proof, and give some typical examples to explain. The convergence of series is prerequisite to the summation, but the focus of this paper is to discuss the method of summation of series, the series convergence discussion over does not talk, and the series is mentioned in this article are convergent.

Key words: Series ConvergenceA number of series summation Methods and skillsSeeking and split method Summing a series of powers

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目录

摘要 (Ⅰ)

Abstract (Ⅱ)

第一章绪论 (1)

1.1 课题的研究背景及意义 (1)

1.2 课题的发展概况及现状 (1)

1.3 全文研究内容及章节安排 (2)

第二章数项级数求和的常用方法 (3)

2.1 数项级数的定义及收敛 (3)

2.2 等差级数求和 (3)

2.3 首尾相加求和 (3)

2.4 等比级数求和 (4)

2.5错位相减求和 (4)

2.6 蕴含型级数相消法求和 (4)

2.7 有理化法求和 (5)

2.8原级数转化为子序列求和 (5)

2.9数项级数化为函数项级数求和 (6)

2.10数项级数化为积分函数求和 (6)

2.11裂项法求和 (6)

2.12夹逼法求和 (7)

第三章关于数项级数求和的几种特殊方法 (9)

3.1 方程式法求和 (9)

3.2 三角型数项级数化为复数系级数求和 (9)

3.3幂级数求和 (10)

3.3.1 利用幂级数的性质求级数和 (10)

3.3.2 利用微分方程的转化求级数和 (11)

3.4利用傅里叶系数求和 (12)

第四章小结 (14)

致谢 (15)

参考文献 (16)

原创性声明 (17)

论文使用授权声明 (17)