关于正项级数收敛性的判别法

关于正项级数收敛性的判别法

On convergence of series with positive terms

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摘要

正项级数作为级数理论中最基本的一类级数，它的敛散性的判定是级数理论的核心问题。正项级数的敛散性判别方法有很多，本文对正项级数敛散性的各种判别法的特点与联系作了简单、系统的归纳与剖析。正项级数不仅有一般级数收敛性的判别法，也有许多常用的和一些新的收敛性的判定方法，如比较判别法、柯西判别法、达朗贝尔判别法、拉贝判别法和对数判别法等，但运用起来有一定的技巧，需要根据对不同级数通项的特点进行分析，选择适宜的方法进行判定，这样才能够最大限度的节约时间，提高效率，特别是对于一些典型问题，运用典型方法，更能事半功倍。

关键词：级数；正项级数；收敛；发散。

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Abstract

Determining whether or not a series is convergent in the series theory is the core issue. There are many ways to determine if a positive series is convergent. This thesis makes full analysis for the convergence determination methods for positive series. There are many common and some new convergence determination methods, such as comparison criterion, Cauchy criterion, d'Alembert criterion, Log Criterion and Rabe Criterion and other methods. But using which of these methods needs certain skills, needs to analyze the general items of the series. A lot of time can be saved if an appropriate method is used.

Key words: Series; positive series; convergence; divergence.

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