交错级数敛散性判别法

给出了交错级数的一个判别法,应用此判别法可直接判别交错级数是否收敛,以及收敛时是绝对收敛还是条件收敛.     

交错级数敛散性的微分形式判别法

给出交错级数敛散性微分形式的判别法,应用此判别法可直接判别交错级数是否收敛,以及收敛时是绝对收敛还是条件收敛.     

基于p-级数的交错级数敛散性判别法

选择p-级数作为参照级数,由比较判别法可得关于交错级数敛散性判别的一种新方法.新方法可直接判别交错级数的敛散性,并在收敛时,给出级数是条件收敛还是绝对收敛.实例说明其应用.     

交错级数敛散性判别法

给出交错级数的几个判别法,它们可直接用以判别交错级数是绝对收敛,条件收敛还是发散.     

几类不满足莱布尼茨判别法条件但收敛的交错级数

本文构建几类不满足莱布尼茨判别法条件但仍收敛的交错级数.     

关于交错级数收敛性判定的探讨

对交错级数的收敛性判定思路进行探讨．运用莱布尼兹判别法结合级数收敛的性质,方便地解决了几种典型交错级数收敛性判定问题．     

交错级数收敛性的几个结果及其应用

莱布尼兹判别法只是一个充分条件。有大量交错级数虽然不满足其条件,但却是收敛的．对于无法用莱布尼兹判别法判定的三类交错级数,利用常数项级数收敛的定义及相关结果,可以证明在一定条件下它们都是收敛的．并通过实例说明所得结果的应用价值．     

交错级数莱布尼兹判别准则的推广

本文给出了交错级数收敛的一个充要条件，推广了交错级数莱布尼兹判别准则。     

广义p-拉贝判别法及其应用

在广义拉贝判别法的基础上,给出了广义p-拉贝判别法及其极限形式.将其应用于判定交错级数的绝对收敛或条件收敛,得到并证明了相关定理.最后,通过若干例子验证了方法的有效性.     

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…     

On Convergence of Double Series

For the convergence of double series and iterated series, a sufficient condition is obtained. This result provides a test for the convergence of double series and iterated series.     

Fourier series approximation of separable models

The approximation of a function affected by noise in several dimensions suffers from the so-called “curse of dimensionality”. In this paper a Fourier series method based on regularization is developed both for uniform and random design when a restriction on the complexity of the curve such as additivity is considered in order to circumvent the problem. Optimal convergence theorems are stated and numerical experiments are shown on several test problems available in the literature together with comparisons with alternative methods.     

A study on the convergence of homotopy analysis method

In this study, a reliable approach for convergence of the homotopy analysis method when applied to nonlinear problems is discussed. First, we present an alternative framework of the method which can be used simply and effectively to handle nonlinear problems. Then, mainly, we address the sufficient condition for convergence of the method. The convergence analysis is reliable enough to estimate the maximum absolute truncated error of the homotopy series solution. The analysis is illustrated by investigating the convergence results for some nonlinear differential equations. The study highlights the power of the method.     

Ratio-Like and Recurrence Relation Tests for Convergence of Series

Three tests for convergence of series are given which are wellsuited for the automatic determination of the radius of convergenceof a series. The first test is similar to the standard ratiotest, while the other two tests use a recurrence relation forthe terms of the series. Together, these tests determine theradius of convergence of many series which arise as solutionsto ordinary differential equations. Numerical examples usingthese tests are given.     

Computing the region of convergence for power series in many real variables: A ratio-like test

We give an elementary proof that the region of convergence for a power series in many real variables is a star-convex domain but not, in general, a convex domain. In doing so, we deduce a natural higher-dimensional analog of the so-called ratio test from univariate power series. From the constructive proof of this result, we arrive at a method to approximate the region of convergence up to a desired accuracy. While most results in the literature are for rather specialized classes of multivariate power series, the method devised here is general. As far as applications are concerned, note that while theorems such as the Cauchy-Kowalevski theorem (and its generalizations to many variables) grant the existence of a region of convergence for a multivariate Taylor series to certain PDEs under appropriate restrictions, they do not give the actual region of convergence. The determination of the maximal region of convergence for such a series solution to a PDE is one application of our result.     

有关常数项级数的几个典型例题

通过实例考察常数项级数收敛和发散时一般项的一些特点,并讨论级数不满足比值判别法、根值判别法或莱布尼茨定理的条件时的收敛性问题.     

A quantitative version of the Dirichlet-Jordan test for double Fourier series

For classes of functions with convergent Fourier series, the problem of estimating the rate of convergence has always been of interest. The classical theorem of Dirichlet and Jordan for functions of bounded variation assures the convergence of their Fourier series, but gives no estimate of the rate of convergence. Such an estimate was first provided by Bojani . Here we consider this problem in the case of functions of two variables that are of bounded variation in the sense of Hardy and Krause. The Dirichlet-Jordan test was first extended by Hardy from single to double Fourier series. Now, we provide a quantitative version of it. We prove our estimate in a greater generality, by introducing the so-called rectangular oscillation of a function of two variables over a rectangle.     

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 .     

The new test criterion for the homogeneity of parameters of several populations

Summary The new test criterion for testing the homogeneity of parameters of several populations is proposed and the test properties of it is discussed. The asymptotic expansions of the distributions of test criterion are discussed under (i) null hypothesis, (ii) fixed alternative hypothesis and (iii) local alternative hypothesis converging to the null hypothesis with appropriate rate of convergence as the sample size increases. As a particular case the asymptotic theory of a statistic for a homogeneity of variances of normal populations is also discussed and the exact moments of it under a null hypothesis can be used to obtain a percentage point by a Pearsonian curve fitting. This Institute of statistical Mathematics