Cauchy判别法的推广

以正项级数的比较判别法为基础,得到判别正项级数敛散性的两个判别方法,它可以作为Cauchy判别法对正项级数∑u_n,(u_n0),ρ=(?)u_n~(n/1)当ρ=1失效时的一个补充,把它称为Cauchy判别法的推广.     

关于Cauchy—D‘Alembert判别法

直接使用Cauchy判别法或者D'alembert判别法来判别数项级数的敛散性时,有时计算极限难度大.为了计算极限简单,本文提出灵活使用Cauchy判别法和D'alembert判别法的方法.     

正项函数项级数一致收敛的Raabe型判别法的推广

证明了正项函数项级数一致收敛的比值判别法,进而得到了正项函数项级数一致收敛的Raabe型判别法,并给出了应用的实例.     

正项级数收敛性的一种新的判别法

将正项级数收敛性的 D′Alembert比值判别法和 Cauchy根值判法的数学思想融合到一起 ,利用正项级数的比较判别法和级数的某些基本性质 ,给出了正项级数收敛性的一种新的判别法 ,暂时称之为 Z-判别法 .     

拟Raabe判别法与拟对数判别法的强弱关系

拟Raabe判别法是新近提出的关于正项级数收敛性的一种比较细致的判别法.对通项递减的正项级数来说,此判别法强于传统的Raabe判别法与Gauss判别法.通过对拟Raabe判别法与另一个细致的判别法——拟对数判别法强弱关系的探讨,得出了后一判别法强于前者的结论.     

数项级数敛散性的判别程序与正项级数的地位和作用

作者曾给出过数项级数敛散性的判别程序,本文对原有框图进行了修改和补充．从框图中不仅可以了解到级数收敛的定义,级数收敛的必要条件、交错级数的莱布尼兹定理以及绝对收敛与收敛的关系,更能体会到正项级数在数项级数中的重要地位．事实上,对一般的级数,如果用正项级数的比值或根值审敛法判定收敛,则收敛;若发散,则发散（只要注意到比值或根值审敛法的证明过程就不难推出这一点）．正是由于这个原因,正项级数在函数项级数的研究中起着十分重要的作用．一、数项级数敛散性的判别程序二、止坝级数在由数坝线教甲同作用众所周知,定…     

正项级数Kummer判别法的一个推广及其应用

本文给出并证明了正项级数 Kum mer判别法的一个推广 ,在更一般的意义下讨论了通常的正项级数判别法 ,扩大了原来的判别法判断敛散性的范围 .     

交错级数收敛性判别法

研究交错级数收敛性判别法.通过计算级数通项的极限和单调性得到三个判据,并对其中两个结论给出形式简化的推论,最后举例说明所提判别法的应用.     

对正项级数敛散性判别方法的研究

利用函数的泰勒展开及极限的运算性质,借助已知敛散性的级数■和■,推出了判别正项级数敛散性的两个方法,并在此基础上得到了通项递减的正项级数敛散性的两个判别法.文中的结论强于双比值判别法.     

正项级数的D—C判别法

本将正项级数的比值审敛法（达朗贝尔D'Alembert判别法）和根值审敛法（柯西Cauchy判别法）结合起来，得到正项级数的一个新的审敛法，且称之为D－C判别法。     

An infinite series involving inverse tan

In this note we introduce an infinite series which represents an interesting challenge for students with the relevant background.     

无限级Dirichlet级数及随机Dirichlet级数

主要研究全平面上无限级Ｄｉｒｉｃｈｌｅｔ级数及随机Ｄｉｒｉｃｈｌｅｔ级数的增长性．对于 Ｄｉｒｉｃｈｌｅｔ级数,研究了它的增长性和正则增长性,得到了它的系数和指数与增长级的两 个充要条件．对于随机Ｄｉｒｉｃｈｌｅｔ级数,证明了它的增长性几乎必然与其在每条水平直线 上的增长性相同．     

P-adic数域Qp上的级数理论

主要研究了P-adic数域上无穷级数的收敛问题,给出了P-adic数域上数项级数,函数项级数收敛的充要条件,并作了完整的证明.得到了P-adic数域上级数收敛比实数域上级数收敛更容易判断的结论.     

Multidimensional Matrix Inversions and Ar and Dr Basic Hypergeometric Series

We compute the inverse of a specific infinite r-dimensional matrix, thus unifying multidimensional matrix inversions recently found by Milne, Lilly, and Bhatnagar. Our inversion is an r-dimensional extension of a matrix inversion previously found by Krattenthaler. We also compute the inverse of another infinite r-dimensional matrix. As applications of our matrix inversions, we derive new summation formulas for multidimensional basic hypergeometric series.     

无穷级数引入的教学设计

给出针对无穷级数引入的一种教学设计.以趣味故事作为实例引入,通过对π的计算这个实际问题的讲解,让学生抓住概念的本质,并对无穷级数的应用有一个初步的了解,从而对无穷级数有一个总体的认识.在教学过程中,步步引导,全程既具有趣味性又具有启发性.     

ON THE GROWTH OF INFINITE ORDER DIRICHLET SERIES

In this paper, the property of infinite order Dirichlet series in the half-plane are investigated. The more exact growth of infinite order Dirichlet series is obtained without using logarithm argument to the type-function for the first time.     

Computation of an infinite integral using series acceleration

A numerical computation in crystallography involves an infinite integral depending on one parameter. In a recent article in this journal this computational problem is addressed using Romberg’s method and tools for error control. One observe numerical difficulties with the reported approach both near the parameter’s endpoints and near the parameter interval’s midpoint. In this short note we will present an alternative approach making use of a known infinite series formulation of the problem at hand and a simple and efficient series acceleration technique. If some care is taken to avoid cancellations the numerical results are excellent for all values of the parameter. AMS subject classification 65B05, 65B10, 65D30     

计数测度在无穷级数中的一个应用

通过引入计数测度,将数学分析中的无穷级数和测度论中的抽象积分联系起来,并在此基础上对双重连加号中,连加号的次序可以颠倒这个性质给出了一个证明.     

Extension on absolute summability factors of infinite series

Extension and improvement has been made on a recent result proved by Mazhar [S.M. Mazhar, Absolute summability factors of infinite series, Kyungpook Math. J. 39 (1999) 67-73] concerning absolute summability factors of infinite series.     

Factorization of matrix functions and their inverses via power product expansions

The conversion of a power series with matrix coefficients into an infinite product of certain elementary matrix factors is studied. The expansion of a power series with matrix coefficients as the inverse of an infinite product of elementary factors is also analyzed. Each elementary factor is the sum of the identity matrix and a certain matrix coefficient multiplied by a certain power of the variable. The two expansions provide us with representations of a matrix function and its inverse by infinite products of elementary factors. Estimates on the domain of convergence of the infinite products are given.