分式线性递推数列极限的案例分析

针对分式线性递推数列,借助具体案例,探讨利用通项求极限、存在性求极限以及数学实验观察极限等多种方法,以期拓展学生的视野和提高学生学习数列极限的积极性。     

利用不动点方法求递推数列的极限

求递推数列的极限是数列极限中一个非常重要的内容,常用单调有界定理,压缩映射原理解决.本文利用不动点给出该类数列的解法,在解决复杂问题中有一定的优越性.     

一道例题的引申与启发

通过对一道关于递推数列求极限例题的引申和探索,说明教师可以启发和引导学生进行变换、推广和类比,使其有所新发现．     

递推数列极限的初等求法和收敛渐近性

借助实例介绍一些非线性递推数列，特别是分式线性递推数列极限的初等求法。就一般分式线性递推数列，明确其收敛渐近性，并通过相关推论展示其应用。     

用解递推关系法求数列的极限

本文给出两个递推关系的求解公式,对某些递推关系通过变换化为可求通项的递推关系式,从而求出极限。如果数列的通项已知,那么,其极限就比较容易求得.而对于象由递推关系等所确定的数列,一般《高等数学》教材上,大多采用诸如单调有界有极限的原理以及级数理论等方法.但有时证明极限存在比较困难,即使假定极限存在,要求出来也并不容易。工科院校学生的数学基础理论一般比较薄弱,对求解此类极限往往不易掌握。而实际上有些由递推关系确定的数列的极限是有简便方法可寻的。本文给出两个公式,对于某些递推关系的通项的求解显得非常简单。     

递归数列极限的一种直接求法

通常,解决递归数列问题的关键在于求得通项公式 a_n,但对于某些特殊的递归数列如只要求出(?)a_n,则完全可以直接从递推公式两边求极限而得到,不必化那么大的精力去求通项公式 a_n.解法如下.     

二阶线型递推数列通项公式的求法

求递推数列的通项公式，既是中学数学学习中的一个难点，又是近几年高考的一个热点，近三年新课程高考压轴题都是求这类数列通项公式的问题．文[1]介绍了一些常见递推数列通项公式的求法，本文就求二阶线型递推数列通项公式，介绍一种通用的方法．     

递推数列通项公式常用求解方法探讨

<正>数列知识是高考中的重点内容,也是必考内容,其中递推数列是数列问题的重中之重.掌握求递推数列的通项公式的转化方法与规律,对于解决数列通项公式问题具有重要作用.由递推数列求通项,形式多变、解法灵活、技巧性强,解法的关键是将递推关系式转化为我们熟知的等差型、等比型、累加型、累乘型等数列形式,然后求出数列的通项公式.此类问题主要有以下     

用不动点求两类递推数列的通项

<正>求递推数列通项是高考以及数学竞赛的重要考点,尤其是在数学竞赛中,数列的递推形式丰富多样,这为求解通项带来一定的难度.利用函数不动点的方法,把递推数列转化为等差、等比或其它方便求通项的递推形式,问题便事半功倍了.本文介绍了利用函数不动点法在复数范围内求解二阶递推数列a_(n+2)=     

一类数列极限的矩阵解法

通过矩阵方法可求一类由常系数线性递推公式所确定的数列的极限.实例演示其递推公式形如xn 1=pxn qxn-1(p,q为非零常数)和xn 1=caxxnn db(c≠0,且ad≠bc)的两类数列{xn}的极限的求法.     

On Discrete Limits of Sequences of Approximately Continuous Functions and tae-Continuous Functions

We show that the classes of all discrete limits of sequences of ap- proximately continuous functions, of all discrete limits of sequences of derivatives and of all discrete limits of sequences of Baire 1 functions are the same. We describe also the discrete limits of sequences of quasicontinuous functions, and of sequences of almost everywhere continuous functions, and we present anec- essary condition which must be satisfied by the discrete limits of sequences of T_ae -continuous functions.     

On Almost Regular Convergence

Regular convergence of multiple sequences, introduced by G. H. Hardy and F. Móricz, can be generalized to almost convergent sequences in various ways. In the paper, classes of almost convergent double sequences with a kind of uniform regularity are studied, which make these classes similar to the class of regularly convergent sequences. Matrices, which convert sequences from these classes to bounded and convergent double sequences with limits equal to the generalized limits of the original sequences are characterized. The results extend results of F. Móricz and B. E. Rhoades on strongly regular matrices.     

Collapsing sequences of solutions to the Ricci flow on 3-manifolds with almost nonnegative curvature

We study sequences of 3-dimensional solutions to the Ricci flow with almost nonnegative sectional curvatures and diameters tending to infinity. Such sequences may arise from the limits of dilations about singularities of Type IIb. In particular, we study the case when the sequence collapses, which may occur when dilating about infinite time singularities. In this case we classify the possible Gromov-Hausdorff limits and construct 2-dimensional virtual limits. The virtual limits are constructed using Fukaya theory of the limits of local covers. We then show that the virtual limit arising from appropriate dilations of a Type IIb singularity is always Hamilton's cigar soliton solution. Partially supported by NSF grant DMS-0203926.     

Rademacher's infinite partial fraction conjecture is (almost certainly) false

In his book Topics in Analytic Number Theory, Hans Rademacher conjectured that the limits of certain sequences of coefficients that arise in the ordinary partial fraction decomposition of the generating function for partitions of integers into at most N parts exist and equal particular values that he specified. Despite being open for nearly four decades, little progress has been made towards proving or disproving the conjecture, perhaps in part due to the difficulty in actually computing the coefficients in question. In this paper, we present a recurrence (alias difference equation) which provides a fast algorithm for calculating the Rademacher coefficients, a large amount of data, direct formulae for certain collections of Rademacher coefficients, and overwhelming evidence against the truth of the conjecture. While the limits of the sequences of Rademacher coefficients do not exist (the sequences oscillate and attain arbitrarily large positive and negative values), the sequences do get very close to Rademacher's conjectured limits for certain (predictable) indices in the sequences.     

On dimension of inverse limits with upper semicontinuous set-valued bonding functions

We give results about the dimension of continua, obtained by combining inverse limits of inverse sequences of metric spaces and one-valued bonding maps with inverse limits of inverse sequences of metric spaces and upper semicontinuous set-valued bonding functions, by standard procedure introduced in [I. Bani?, Continua with kernels, Houston J. Math. (2006), in press].     

两两NQD序列与随机Dirichlet级数

本文研究了一类两两NQD序列加权乘积和的Jamison型强稳定性.利用两两NQD序列的一些极限理论,获得了系数的模为两两NQD列的一类随机Difichlet级数的增长性和值分布的一些新的结果,推广和改进了两两独立序列的相关结论.     

两个数列极限的推广

证明了四个命题 ,它们是两个已知的数列极限的推广 .     

两个数列极限的再推广

证明了三个命题,它们是已知的数列极限的推广.     

Hyperfinite knots via the CJKLS invariant in the thermodynamic limit

We set forth a definition of hyperfinite knots. Loosely speaking, these are limits of certain sequences of knots with increasing crossing number. These limits exist in appropriate closures of quotient spaces of knots. We give examples of hyperfinite knots. These examples stem from an application of the Thermodynamic Limit to the CJKLS invariant of knots.