无穷积分敛散性的一些判别方法及其运用.doc

摘 要 微积分知识及其应用对人类的进步和社会的发展做出了突出贡献, 无穷积分在其中也发挥了重要作用. 无穷积分在生活中常见且应用十分广泛. 本文首先介绍了无穷积分收敛与发散的相关概念，然后对判断无穷积分的敛散性的方法做了详细的归纳和总结. 本文主要对无穷积分的敛散性判别方法进行了详细的介绍. 从中得到不同的被积函数可以有不同的判别方法，其一，被积函数是非负函数的情况下无穷积分是否收敛，我们可以应用定义判别法、柯西准则、比较判别法、根值判别法来判别无穷积分的敛散性. 其二，在通常的条件下无穷积分敛散性的判别，我们一般考虑使用狄利克雷判别法、阿贝尔判别法、对数判别法来解决. 最后通过一些实例研究了无穷积分在物理学、经济学等实际问题的应用. 关键词：无穷积分；收敛；发散；绝对收敛 Abstract Calculus knowledge and its application have made outstanding contributions to human progress and social development, and infinite integral has also played an important role in it. Infinite integral is common in life and widely used. This paper first introduces the related concepts of convergence and divergence of infinite integral, and then to determine the infinite integral convergence of the method in detail. This paper focuses on the convergence of infinite integral method is introduced in detail. The different integrand can have different discriminant method on the one hand, the integrand is nonnegative function under the condition of infinite integral is convergent, we can define the application criterion, Cauchy criterion, comparative method, value method to determine the convergence of infinite integral. Secondly, under normal conditions the convergence and divergence of infinite integral, we generally consider the use of de Lickley criterion, Abel criterion and logarithmic discriminant method to solve the problem. Finally, through the application of some examples of the infinite integral in physics, economics and other practical problems. Key words: Infinite Integral; Convergence; Divergence; Absolute Convergence 不要删除行尾的分节符，此行不会被打印目 录 摘要 I Abstract II 绪论 1 第1章 相关概念知识 3 1.1 预备知识 3 1.2 无穷积分敛散性的概念 3 1.2.1 无穷积分收敛与发散 3 1.2.2 无穷积分与级数 5 1.2.3 无穷积分的性质 6 1.2.4 绝对收敛与条件收敛 6 第2章 敛散性的判别方法 7 2.1 被积函数是非负的判别 7 2.1.1 定义判别法 7 2.1.2 柯西准则 7 2.1.3 比较判别法 8 2.1.4 根值判别法 9 2.2 一般情况下的判别 11 2.2.1 狄利克雷判别法 11 2.2.2 阿贝尔判别法 12 2.2.