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用放缩法证明不等式是一种重要的常规方法,强化对这种证明方法的训练,可有效地提升数学解题能力。下面笔者结合具体例题进行分析。1放缩代表项如在所证明的不等式中含有某数列的和式,常可将其代表项(即通项)进行放缩,然后再进行证明。例1求证:1/(1~2)+1/(2~2)+1/(3~2)+…+1/(k~2)<2(k∈N)。证明:先对数列的代表项1/n~2进行适度的放缩,即有1/(n~2)<1/n(n-1)=1/(n-1)-1/n(n≥2)。①将n=2,3,…,k代人得1/(1~2)=1,1/(2~2)<1/1-1/2,1/(3~2)<1/2-1/3,…,1/(k~2)<1/(k-1)-1/k。
Proving inequality by means of scaling is an important conventional method. Strengthening the training of this proof method can effectively improve the ability of solving math problems. The following combination of specific examples for analysis. 1 Shrink representation If it is proved that the inequality contains a series of sums, often its representative (that is, general terms) scaling, and then prove. Example 1 Verify that: 1 / (1 ~ 2) + 1 / (2 ~ 2) + 1 / (3 ~ 2) + ... + 1 / (k ~ 2) <2 (k∈N). Proof: First, the representative items 1 / n ~ 2 of the series are moderated and scaled, that is, 1 / (n ~ 2) <1 / n (n-1) = 1 / (n-1) -1 / n n ≧ 2). 1) n = 2, 3, ..., k generations have 1 / (1 to 2) = 1, 1 / (2 to 2) 2-1 / 3, ..., 1 / (k ~ 2) <1 / (k-1) -1 / k.