迭代数列的敛散性及其相关应用  

Convergence and Divergence of Iterative Sequence and Its Related Applications

作　　者：马东阳 王可 叶润萍[1] Ma Dongyang;Wang ke;Ye Runping(College of Letters and Science,Suqian College,Suqian 223800,China)

机构地区：[1]宿迁学院文理学院,江苏宿迁223800

出　　处：《科学咨询》2022年第7期57-60,共4页

摘　　要：为研究用递推公式生成的迭代数列的敛散性,常用教材上给的单调有界原理进行验证,证明有界性常用的方法有:从递归关系式观察得出;用已知不等式推出;用归纳法证明等,证明单调性常用的方法有:比值法、差值法、利用递推式判断函数的单调性等。但是,经常会发现有些给定的数列不具有单调性,此时可以利用单调有界原理判定奇数列和偶数列的极限,若两极限相等,则原数列收敛。此外,本文还将介绍利用压缩映射原理、不动点方法、上下极限法来判断迭代数列的收敛性,并介绍迭代数列在方程求根与近似计算方面的应用。In order to study the convergence and divergence of the iterative sequence generated by recursive formula,the monotone bounded principle given in the teaching materials is usually used as a proof.The commonly used methods to prove boundedness are:from the observation of recursive relation;deriving from the known inequality;proving by induction and etc.The methods to prove monotonicity are:ratio method,difference method,and monotonicity of function by recursive judgment and so on.However,it is often found that some given series do not have monotonicity.At this time,the limit of odd and even columns can be determined by monotone bounded principle.If the two limits are equal,the original series converges.In addition,this paper will introduce the convergence of iterative sequence using compression mapping principle,fixed point method and upper and lower limit method,and introduce the application of iterative sequence in equation root finding and approximate calculation.

关 键 词：迭代数列 敛散性 近似计算 

分 类 号：G63[文化科学—教育学]