设P1,P2···,Pn是1,2,···,n的任意排列求证:1/(P1+P2)+1/(P2+P3)+···+1/(Pn-1+Pn)>(n-1)/(n+2)大手来解.过程要看的懂啊.
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![设P1,P2···,Pn是1,2,···,n的任意排列求证:1/(P1+P2)+1/(P2+P3)+···+1/(Pn-1+Pn)>(n-1)/(n+2)大手来解.过程要看的懂啊.](/uploads/image/z/5005974-30-4.jpg?t=%E8%AE%BEP1%2CP2%C2%B7%C2%B7%C2%B7%2CPn%E6%98%AF1%2C2%2C%C2%B7%C2%B7%C2%B7%2Cn%E7%9A%84%E4%BB%BB%E6%84%8F%E6%8E%92%E5%88%97%E6%B1%82%E8%AF%81%3A1%2F%28P1%2BP2%29%2B1%2F%28P2%2BP3%29%2B%C2%B7%C2%B7%C2%B7%2B1%2F%28Pn-1%2BPn%29%3E%28n-1%29%2F%28n%2B2%29%E5%A4%A7%E6%89%8B%E6%9D%A5%E8%A7%A3.%E8%BF%87%E7%A8%8B%E8%A6%81%E7%9C%8B%E7%9A%84%E6%87%82%E5%95%8A.)
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设P1,P2···,Pn是1,2,···,n的任意排列求证:1/(P1+P2)+1/(P2+P3)+···+1/(Pn-1+Pn)>(n-1)/(n+2)大手来解.过程要看的懂啊.
设P1,P2···,Pn是1,2,···,n的任意排列求证:1/(P1+P2)+1/(P2+P3)+···+1/(Pn-1+Pn)>(n-1)/(n+2)
大手来解.过程要看的懂啊.
设P1,P2···,Pn是1,2,···,n的任意排列求证:1/(P1+P2)+1/(P2+P3)+···+1/(Pn-1+Pn)>(n-1)/(n+2)大手来解.过程要看的懂啊.
用Cauchy不等式.
((P1+P2)+(P2+P3)+...+(P(n-1)+Pn))(1/(P1+P2)+1/(P2+P3)+...+1/(P(n-1)+Pn))
≥ (1+1+...+1)² = (n-1)².
而(P1+P2)+(P2+P3)+...+(P(n-1)+Pn) = 2(P1+P2+...+Pn)-P1-Pn.
P1,P2,...,Pn是1,2,...,n的一个排列,故P1+P2+...+Pn = 1+2+...+n = n(n+1)/2.
又P1+Pn ≥ 1+2 > 2,故(P1+P2)+(P2+P3)+...+(P(n-1)+Pn) < 2n(n+1)/2-2 = n²+n-2 = (n-1)(n+2).
于是1/(P1+P2)+1/(P2+P3)+...+1/(P(n-1)+Pn)
≥ (n-1)²/((P1+P2)+(P2+P3)+...+(P(n-1)+Pn)) > (n-1)/(n+2).