裂项相消法的解题策略

Solution to the problem of splitting phase elimination

本文研究裂项相消法的解题策略。首先研究了裂项相消法的实用范围为通项服从特征:an=bnbn+k或an=bn+k-bn;接着给出了如何将不明显具有上述特征的数列,如何经过裂项变形,使之满足条件;最后,我们给出了相消后所剩的项和项数与bn, bn+k项数之差k有关,相消过程中所剩下的项有如下对称的结论:前k面剩k项被减项b_1, b_2,…b_k,后面剩k项减项b_(n-k), b_(n+k-1), b_(n+1)。即:{an}的前n项和S_n=kΣi=1(b_i-b_(n+i))。

This paper studies the problem solving strategy of split phase elimination.Firstly,the practical range of the split-term cancellation method is studied,which is called an=bn-bn+k or an=bn+k-bn.Then,how to distort the sequence which does not have the above characteristics and how to satisfy the conditions are given.Finally,we show that the remaining terms and the number of terms after cancellation are related to the difference K of the number of bn,bn+k terms.The rest of the items have the following symmetry conclusion:the remaining K item is reduced by b1,b2,…bk,and the remaining item minus bn-k,bn+k-1,bn+1.Namely,the former n item and Sn=■(bi-bn+i)of{an}.