关于在求数列极限问题时极限存在准则的应用新解

A New Solution for the Application of the Criterion for the Existence of Limits When Solving the Limit Problem of a Sequence

为得出以逐项公式为条件的求数列极限问题的通法通解,本文从新的角度分别利用单调有界准则和夹逼准则求解一道经典的数列极限问题,从中提炼出求数列极限问题思路方法,将其应用在其他同类型问题上。通过探究,我们得出了“先算极限再证明收敛”的基本解题流程,并进一步得出了两准则各自适用的解题情形。本探究揭示了以逐项公式为条件的求数列极限问题区别于一般以通项公式为条件的求数列极限问题的本质区别,强调了该类型问题与一般求数列极限问题等同的重要性。

In order to obtain the general solution of solving the limit problem of sequence of sequences based on the term by term formula, this paper uses the monotonic bounded criterion and the squeeze criterion respectively to solve a classical limit problem of sequence of sequences from a new angle, and extracts the thought method of solving the limit problem of sequence of sequences, which is applied to other similar problems. Through exploration, we obtained the basic problem solving process of “calculate the limit first and then prove the convergence”, and further obtained the problem solving situation applicable to the two criteria. This research reveals the essential difference between the problem of finding the limit of series based on the term formula and the problem of finding the limit of series based on the general term formula, and emphasizes the im-portance of this type of problem being equal to the general problem of finding the limit of series.