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.     

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．     

一类交错级数的收敛定理

讨论和分析了一类交错级数的收敛问题,给出了异于莱布尼兹判别法的关于交错级数的一个收敛定理.我们的结论还推广了正项级数的拉阿伯判别法的使用范围.     

交错级数的对数判别法

从正项级数的Raabe对数判别法入手,给出了交错级数的一个新的审敛方法.与文[1],[2]所给的审敛法相比,当交错级数的一般项含有幂指项时,利用该审敛法判断其敛散性显得尤为简便.     

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.     

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.     

交错级数收敛性的一个判别法

当交错级数∑∞n=0(-1)n-1un中un含有阶乘、连乘职、幂次等的商的复杂形式时,运用命题中所给出的判别法,判断其收敛性比较方便.     

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.     

关于交错级数收敛的判定法的补充

以交错级数收敛判定法莱布尼兹定理为基础,补充两个结论用来判断某些特殊交错级数的敛散性.     

Discussion on the Leibniz rule and Laplace transform of fractional derivatives using series representation

ABSTRACT

Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. On this basis, the Lebiniz rule and Laplace transform of fractional calculus is investigated. It is analytically shown that the commonly used Leibniz rule cannot be applied for Caputo derivative. Similarly, the well-known Laplace transform of Riemann–Liouville derivative is doubtful for n-th continuously differentiable function. After pointing out such problems, the exact formula of Caputo Leibniz rule and the explanation of Riemann–Liouville Laplace transform are presented. Finally, three illustrative examples are revisited to confirm the obtained results.     

A note on the summation of slowly convergent alternating series

Thekth term of the infinite series is larger than 0.5 wheneverk <k ₀, wherek ₀ + 1 =e ¹⁰²⁴. To sum this series correct to order 10^–1 using direct summation seems an impossible task, notwithstanding the power of modern computers. This note will present an alternative approach to those classical methods (the Euler transformation is one) which can accurately sum such series. The theory to be presented has the added advantage of providing accurate bounds for the error in the approximate result. The method used will be Euler-Maclaurin summation, revitalised by computer algebra.The sum of theintegrated series asx 0.     

Hausdorff dimension of some kind of alternating Oppenheim series expansion

For Oppenheim series epansions, the authors of [7] discussed the exceptional sets Bm={x∈（0,1]：1〈dj（x）/h（j-1）（d（j-1）（x））≤m for any j ≥2} In this paper, we investigate the Hausdorff dimension of a kind of exceptional sets occurring in alternating Oppenheim series expansion. As an application, we get the exact Hausdorff dimension of the-set in Luroth series expansion, also we give an estimate of such dimensional number.     

The Fourier-Walsh series of the functions absolutely continuous in the generalized restricted sense

We consider the questions of convergence in Lorentz spaces for the Fourier-Walsh series of the functions with Denjoy integrable derivative. We prove that a condition on a function f sufficient for its Fourier-Walsh series to converge in the Lorentz spaces “near” L _∞ cannot be expressed in terms of the growth of the derivative f′.     

Characterizing the derivative and the entropy function by the Leibniz rule

Consider an operator T:C¹(R)→C(R) satisfying the Leibniz rule functional equation

     

三类特殊形式的变号级数敛散性判别法

根据Leibniz级数的定义和判别法,定义了三种类Leibniz级数并给出了相应的判别法.     

PAHSS-PTS alternating splitting iterative methods for nonsingular saddle point problems

In this paper, we propose the PAHSS-PTS alternating splitting iterative methods for nonsingular saddle point problems. Convergence properties of the proposed methods are studied and corresponding convergence results are given under some suitable conditions. Numerical experiments are presented to confirm the theoretical results, which impliy that PAHSS-PTS iterative methods are effective and feasible.