函数列一致收敛性及其应用

UNIFORM CONVERGENCE OF FUNCTION SEQUENCE AND ITS APPLICATION

函数列及函数项级数的一致收敛性判定是数学分析中的重点难点,对研究函数列的极限函数的连续性、可导性和可积性等起着重要的作用.首先对二元函数列和二元函数项级数一致收敛的定义及其一致收敛的判定方法进行简单说明,然后运用数形结合的思想方法,借助MATLAB软件将具体函数列直观刻画出来,展示其一致收敛的动态过程,揭示其收敛的本质特征,给出二元函数列和二元函数项级数一致收敛的一种新的判定方法.数学软件的应用,能够帮助学生有效利用数形结合的思想对函数列的一致收敛性问题进行直观地分析.这对判断函数列和函数项级数在特定区域内是否一致收敛具有重要意义.

The uniform convergence of function sequence and function series is the key and difficult point in mathematical analysis.It plays an important role in studying the continuity,differentiability and integrability of limit function of function sequence.Firstly,the definition of uniform convergence of binary function series and the determination method of uniform convergence are explained.Then,with the methods of combination of number and shape,this paper uses MATLAB software to describe the specific function sequence,to show its dynamic process of uniform convergence,to reveal the essential characteristics of its convergence,and to give a new method to judge the uniform convergence of binary function sequence.The application of mathematical software not only can judge the convergence and divergence of function sequence more efficiently,but also can effectively use the idea of combination of number and shape to analyze the uniform convergence of function sequence through the image to find out the law,and then clarify the essential characteristics of uniform convergence.It is of great significance to judge whether the function sequence and function series converge uniformly in a specific region.