1.引言设p_N和P_N分别为单位圆的内接及外切正N边形的周长的一半。那么有p_3=3(3~1/2)/2,P_3=3(3~1/2),p_4=2(2~1/2),P_4=4。从几何上看,显然序列{P_N}及{P_N}分别是单调上升及单调下降序列,具有共同极限π。这就是阿基米德求π的近似值的基本方法(可参看Heath[2].)阿基米德利用初等几何的推理方法,得到了下面的递推关系,在此关系式中这两个序列是交错地联 1. INTRODUCTION Let p_N and P_N be half the circumference of the inner circle of the unit circle and the outer N-edge polygon, respectively. Then there are p_3=3(3~1/2)/2, P_3=3(3~1/2), p_4=2(2~1/2), P_4=4. From a geometrical point of view, it is clear that the sequences {P_N} and {P_N} are monotonically increasing and monotonically decreasing sequences, respectively, with a common limit π. This is the basic method by which Archimedes finds the approximate value of π (see Heath [2].) Archimedes uses the method of elementary geometry inference to obtain the following recurrence relation, in which the two sequences are Is intertwined