高二数列题:设数列{an}满足an+1=an^2-nan+1,n为正整数,当a1>=3时,证明……设数列{an}满足an+1=an^2-nan+1,n为正整数,当a1>=3时,证明(2)1/(1+a1) + 1/(1+a2) + ……+1/(1+an) =< 1/2
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![高二数列题:设数列{an}满足an+1=an^2-nan+1,n为正整数,当a1>=3时,证明……设数列{an}满足an+1=an^2-nan+1,n为正整数,当a1>=3时,证明(2)1/(1+a1) + 1/(1+a2) + ……+1/(1+an) =< 1/2](/uploads/image/z/7138257-33-7.jpg?t=%E9%AB%98%E4%BA%8C%E6%95%B0%E5%88%97%E9%A2%98%EF%BC%9A%E8%AE%BE%E6%95%B0%E5%88%97%7Ban%7D%E6%BB%A1%E8%B6%B3an%2B1%3Dan%5E2-nan%2B1%2Cn%E4%B8%BA%E6%AD%A3%E6%95%B4%E6%95%B0%2C%E5%BD%93a1%3E%3D3%E6%97%B6%2C%E8%AF%81%E6%98%8E%E2%80%A6%E2%80%A6%E8%AE%BE%E6%95%B0%E5%88%97%7Ban%7D%E6%BB%A1%E8%B6%B3an%2B1%3Dan%5E2-nan%2B1%2Cn%E4%B8%BA%E6%AD%A3%E6%95%B4%E6%95%B0%2C%E5%BD%93a1%3E%3D3%E6%97%B6%2C%E8%AF%81%E6%98%8E%282%291%2F%281%2Ba1%29+%2B+1%2F%281%2Ba2%29+%2B+%E2%80%A6%E2%80%A6%2B1%2F%281%2Ban%29+%3D%3C+1%2F2)
高二数列题:设数列{an}满足an+1=an^2-nan+1,n为正整数,当a1>=3时,证明……设数列{an}满足an+1=an^2-nan+1,n为正整数,当a1>=3时,证明(2)1/(1+a1) + 1/(1+a2) + ……+1/(1+an) =< 1/2
高二数列题:设数列{an}满足an+1=an^2-nan+1,n为正整数,当a1>=3时,证明……
设数列{an}满足an+1=an^2-nan+1,n为正整数,当a1>=3时,证明
(2)1/(1+a1) + 1/(1+a2) + ……+1/(1+an) =< 1/2
高二数列题:设数列{an}满足an+1=an^2-nan+1,n为正整数,当a1>=3时,证明……设数列{an}满足an+1=an^2-nan+1,n为正整数,当a1>=3时,证明(2)1/(1+a1) + 1/(1+a2) + ……+1/(1+an) =< 1/2
用1-(1/2+1/4+.+1/2^(n-1)),()中的内容用等比数列求和的方法,然后会发现1更大,你追问的问题成立
因为an>=n+2,所以an-n>=2
A(n+1)=An(An-n)+1>=2An+1
A(n+1)+1>=2(An+1)
1/(A(n+1)+1)<=1/(An+1)*1/2
1/(1+a1)=1/4,1/(1+a2)<=1/2*1/4
1/(1+a1) + 1/(1+a2) + ……+1/(1+an) <=1/4(1+1/2+1/4+.....+1/2...
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因为an>=n+2,所以an-n>=2
A(n+1)=An(An-n)+1>=2An+1
A(n+1)+1>=2(An+1)
1/(A(n+1)+1)<=1/(An+1)*1/2
1/(1+a1)=1/4,1/(1+a2)<=1/2*1/4
1/(1+a1) + 1/(1+a2) + ……+1/(1+an) <=1/4(1+1/2+1/4+.....+1/2^(n-1))<1/4*2=1/2
当且仅当n=1时取等号。
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