已知数列an=n(n+1),bn=(n+1)^2,求证1/(a1+b1)+1/(a2+b2)+1/(a3+b3)+……+
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已知数列an=n(n+1),bn=(n+1)^2,求证1/(a1+b1)+1/(a2+b2)+1/(a3+b3)+……+1/(an+bn)
![已知数列an=n(n+1),bn=(n+1)^2,求证1/(a1+b1)+1/(a2+b2)+1/(a3+b3)+……+](/uploads/image/z/4321284-60-4.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97an%3Dn%28n%2B1%29%2Cbn%3D%28n%2B1%29%5E2%2C%E6%B1%82%E8%AF%811%2F%28a1%2Bb1%29%2B1%2F%28a2%2Bb2%29%2B1%2F%28a3%2Bb3%29%2B%E2%80%A6%E2%80%A6%2B)
(an+bn)=n(n+1)+(n+1)^2
=(n+1)(2n+1)>2n(n+1)
1/(an+bn)=1/(n+1)(2n+1)
=2/(2n+1)-2/(2n+2)
从第二项开始放缩,即1/(an+bn)=1/(n+1)(2n+1)< 1/2n(n+1)=1/2(1/n-1/n+1)
1/(a1+b1)+1/(a2+b2)+1/(a3+b3)+……+1/(an+bn)<
2/3-2/4+1/2(1/2-1/3+1/3-1/4+.+1/n-1/n+1
=1/6+1/2(1/2-1/n+1)
=(n+1)(2n+1)>2n(n+1)
1/(an+bn)=1/(n+1)(2n+1)
=2/(2n+1)-2/(2n+2)
从第二项开始放缩,即1/(an+bn)=1/(n+1)(2n+1)< 1/2n(n+1)=1/2(1/n-1/n+1)
1/(a1+b1)+1/(a2+b2)+1/(a3+b3)+……+1/(an+bn)<
2/3-2/4+1/2(1/2-1/3+1/3-1/4+.+1/n-1/n+1
=1/6+1/2(1/2-1/n+1)
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