由单调函数y=f(x)确定的数列x_(n+1)=f(x_(n))收敛性

Convergence x_(n+1)=f(x_(n)) of Sequence Determined by Monotonic Function y=f(x)

针对一类由函数y=f(x)所确定的递推数列x_(n+1)=f(x_(n)),n=0,1,2,…为研究对象,假设f(x)具有单调性,在没有通项公式的情况下,给出了数列{x_(n)}敛散性的判别方法,并总结了判别方法的一般规律,最后给出具体应用以验证方法的有效性和可行性。

This article takes a type of recursive sequence x_(n+1)=f(x_(n)),n=0,1,2,…,determined by a function y=f(x)as the research object.Assuming that f(x)is monotonic,and in the absence of a general term formula,the method for judging the convergence and divergence of the sequence of{x_(n)} is proposed,and the general law of the judgment method is summarized.Finally,specific applications are given to verify the effectiveness and feasibility of the method.