求证：1/2^3+1/3^3+1/4^3+……+1/(n+1)^3<1/4
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解决时间 2021-01-20 10:55
- 提问者网友：嘚啵嘚啵
- 2021-01-19 12:23
求证：1/2^3+1/3^3+1/4^3+……+1/(n+1)^3<1/4
最佳答案
- 五星知识达人网友：渡鹤影
- 2019-05-01 20:24
1/2^3 +1/3^3 +1/4^3 +……+1/(n+1)^3<
(n+1)³=（n²+2n+1)(n+1)>(n²+2n)(n+1)=n(n+1)(n+2)
1/ (n+1)³<1/n(n+1)(n+2) =0.5*(n+2-n)/n(n+1)(n+2)
=0.5[(n+2)/n(n+1)(n+2)】-0.5【n/n(n+1)(n+2)]=0.5[1/n(n+1)-1/(n+1)(n+2)]
l令n=2,3,4 ………… 相加得：
1/2^3 +1/3^3 +1/4^3 +……+1/(n+1)^3
<0.5[1/1*2-1/2*3+1/2*3-1/3*4+……+1/n(n+1)-1/(n+1)(n+2)]=0.5【1/2-/(n+1)(n+2)]<0.5*1/2=1/4
(n+1)³=（n²+2n+1)(n+1)>(n²+2n)(n+1)=n(n+1)(n+2)
1/ (n+1)³<1/n(n+1)(n+2) =0.5*(n+2-n)/n(n+1)(n+2)
=0.5[(n+2)/n(n+1)(n+2)】-0.5【n/n(n+1)(n+2)]=0.5[1/n(n+1)-1/(n+1)(n+2)]
l令n=2,3,4 ………… 相加得：
1/2^3 +1/3^3 +1/4^3 +……+1/(n+1)^3
<0.5[1/1*2-1/2*3+1/2*3-1/3*4+……+1/n(n+1)-1/(n+1)(n+2)]=0.5【1/2-/(n+1)(n+2)]<0.5*1/2=1/4
全部回答
- 1楼网友：摆渡翁
- 2019-06-19 07:56
1/n(n+1)(n+2)=(1/n(n+1)-1/(n+1)(n+2))*1/2
所以,
1/1*2*3 +1/2*3*4+...+1/n(n+1)(n+2)
=[(1/1*2-1/2*3)+(1/2*3-1/3*4)+...+(1/n(n+1)-1/(n+1)(n+2)]*1/2
=(1/2-1/(n+1)(n+2))*1/2
<1/2*1/2=1/4
所以,
1/1*2*3 +1/2*3*4+...+1/n(n+1)(n+2)＜1/4
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