我们知道,柯西不等式:ai,bi∈R,则sum from i=1 to n ai2 sum from i=1 to n bi2≥(sum from i=1 to n aibi)2……(1)当且仅当ai=kbi(i=1,2,…,n)不等式等号成立。它可以作如下变形: 由(1)得(sum from i=1 to n ai2 sum from i=1 to n bi2)1/2≥sum from i=1 to n aibi,添项变为sum from i=1 to n ai2+2 (sum from i=1 to n ai2 sum from i=1 to n bi2)1/2+sum from i=1 to n bi2≥sum from i=1 to n ai2+2sum from i=1 to n aibi+sum from i=1 to n bi2,或sum from i=1 to n ai2-2 (sum from i=1 to n ai2 sum from i=1 to n bi2)1/2+sum from i=1 to n bi2≤sum from i=1 to n ai2-2 sum from i=1 to n ai bi+sum from i=1 to n bi2,分别配方,并开方转
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