逐项微分和逐项积分后的幂级数在收敛区间端点处的审敛法

本文介绍逐项微分和逐项积分后的幂级数在收敛区间端点处的敛散性判定。     

有关幂级数在收敛区间端点上的性质的一些讨论

<正> 对于幂级数的分析运算,同济大学数学教研室主编的《高等数学》下册(1982年第二版)在叙述了幂级数经逐项求导或逐项积分后所得到的幂积数与原幂积数有相同的收敛半径R 以后,在250页有这样一段:     

关于幂级数收敛半径求法的注记

幂级数是微积分应用的重要理论基础,其中收敛半径的求法是学习相关内容的重点和难点.面向工科的高等数学教学中,通常限于介绍求比较简单的幂级数的收敛半径的方法,对于一般的幂级数,由于涉及上极限的理论,高等数学中不做讨论.本文从有界的角度讨论幂级数的收敛半径问题,避开了上极限问题的困难,所得结果可用于求任意幂级数的收敛半径.     

用代数方法求幂级数的和函数

所谓用代数方法求幂级数的和函数是指仅用幂级数的加、减运算及已知的基本展开式来求幂级数在收敛区间内的和函数．有时,用这种方法比用逐项微分、逐项积分更简单、有效．先看一个简单的情形．命题一设数列是公差为d的等差数列,则对应幂级数的和函数为证由比值法容易求得这个幂级数的收敛半径两边同乘,得由于数列入是等差数列,即,故例1在收敛区间内,求幂级数的和函数．解。则幂级数变形为它的系数构成公差为的等差数列,,于是由（l）式得利用（l）式及命题一的证明方法,还能解决相邻两项系数之差构成等差数列的幂级数的求和问题．例…     

幂级数求和函数的方法及应用

幂级数求和函数是无穷级数问题中的重点和难点,该文针对幂级数求和函数总结出其常见类型和解法,求和函数时需要注意的几个问题,以及幂级数求和函数在级数求和、求极限等方面的应用.     

抽象幂级数收敛半径的若干求法

本文依据公式法、幂级数和函数的性质、柯西乘积的结论给出了若干幂级数收敛半径的求法.     

幂级数收敛半径的一些求法

讨论了在一些特殊情形下 ,幂级数收敛半径的求法     

关于幂级数收敛域的几点注记

<正> 对幂级数进行代数运算后所得到的仍然是幂级数,这些运算在幂级数的研究及应用中经常碰到。而每给出一个幂级数都应同时指出它的收敛域,因此,正确地确定经代数运算后     

收敛幂级数环的一个性质

本文证明了收敛幂级数环的张量积不是Ｎｏｅｔｈｅｒ环，并提出一个问题。     

求幂级数收敛半径的方法

指出了文 [1 ]中一个考研题的错误解法 ,并给出求幂级数收敛半径的几种方法     

Convergence for Fourier series solutions of the forced harmonic oscillator II

This paper compliments two recent articles by the author in this journal concerning solving the forced harmonic oscillator equation when the forcing is periodic. The idea is to replace the forcing function by its Fourier series and solve the differential equation term-by-term. Herein the convergence of such series solutions is investigated when the forcing function is bounded, piecewise continuous, and piecewise smooth. The series solution and its term-by-term derivative converge uniformly over the entire real line. The term-by-term differentiation produces a series for the second derivative that converges pointwise and uniformly over any interval not containing a jump discontinuity of the forcing function.     

Convergence for Fourier series solutions of the forced harmonic oscillator

Harmonic oscillator equations of the form ÿ + ?²y = h(t) where ? is a real constant and h(t) is a continuous, piecewise smooth, periodic ‘forcing’ function are considered. The exact solution, obtained through the Laplace transform is cumbersome to handle over long t intervals, and thus solving ‘term-by-term’ by replacing h(t) by its Fourier series is an attractive and accurate alternative. But this solution is an infinite series involving sums of sine and cosine terms, and thus one should worry about convergence of a solution in this form. In the article, it is shown that such a series solution indeed converges uniformly over the entire real line and is twice continuously differentiable, the derivatives being calculated ‘term-by-term’. Only results commonly available in the undergraduate literature are used to verify this and in so doing, a non-trivial application of these results is given. Also included are some interesting problems suitable for undergraduate research.     

Asymptotic expansions of integrals: The term-by-term integration method

The classical term-by-term integration technique used for obtaining asymptotic expansions of integrals requires the integrand to have an uniform asymptotic expansion in the integration variable. A modification of this method is presented in which the uniformity requirement is substituted by a much weaker condition. As we show in some examples, the relaxation of the uniformity condition provides the term-by-term integration technique a large range of applicability. As a consequence of this generality, Watson's lemma and the integration by parts technique applied to Laplace's and a special family of Fourier's transforms become corollaries of the term-by-term integration method.     

The convergence of interpolatory product integration rules

Conditions are given for the convergence of product integration rules based on the zeros of orthogonal polynomials associated with a generalized smooth Jacobi weight, possibly augmented by one or both endpoints.     

The speed of convergence in the Ergodic Theorem

No general statement can be made about the speed with which convergence takes place in the Ergodic Theorem, in the sense that one can never be sure that convergence of the remainder to zero is fast enough to make, through term-by-term multiplication, any divergent series become convergent. As a corollary we obtain the nonexistence in general of the one-sided ergodic Hilbert transform; that is, there is no pointwise nonhomogeneous ergodic theorem.     

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.     

n阶变系数线性差分方程的解

本文利用变数算符￣［２］以及给出变数算符和移动算符的乘积关系，并定义变系数移动算符幂级数间的乘积且证明其在Ｍｉｋｕｉｕｓｋｉ收敛意义下是正确的；另外，把一般的ｎ阶变系数线性差分方程转化为一个恰当的算符方程组，从而获得一般ｎ阶变系数线性差分方程的解。     

Mixed problem for a first-order partial differential equation with involution and periodic boundary conditions

The Fourier method is used to find a classical solution of the mixed problem for a first-order differential equation with involution and periodic boundary conditions. The application of the Fourier method is substantiated using refined asymptotic formulas obtained for the eigenvalues and eigenfunctions of the corresponding spectral problem. The Fourier series representing the formal solution is transformed using certain techniques, and the possibility of its term-by-term differentiation is proved. Minimal requirements are imposed on the initial data of the problem.     

The Convergence of Infinite Element Method for the Non-Similar Case

We have considered the infinite element method for a class of elliptic systems with constant coefficients in [1]. This class can be characterized as: they have the invariance under similarity transformations of independent variables. For example, the Laplace equation and the system of plane elastic equations have this property. We have suggested a technique to solve these problems by applying this property and a self similar discretization, and proved the convergence. Not only the average convergence of the solutions has been discussed, but also term-by-term convergence for the expansions of the solutions. The second convergence manifests the advantage of the infinite element method, that is, the local singularity of the solutions can be calculated with high precision.