问题: 分式
试证:对任意的正整数n，有1/1*2*3+1/2*3*4+...+1/n(n+1)(n+2)<1/4
解答:
1/1*2*3+1/2*3*4+……1/n(n+1)(n+2)
=1/2(1/1*2-1/2*3)+1/2(1/2*3-1/3*4)+...+1/2[1/n(n+1)-1/(n+1)(n+2)]
=1/2[1/1*2-1/2*3+1/2*3-1/3*4+...+1/n(n+1)-1/(n+1)(n+2)]
=1/2[1/1*2-1/(n+1)(n+2)]
=1/2*[(n+1)(n+2)-2]/2(n+1)(n+2)
=(n^2+3n)/4(n+1)(n+2)
=n(n+3)/[4(n+1)(n+2)]
=0.25*n(n+3)/[(n+1)(n+2)]
n(n+3)/[(n+1)(n+2)]=(n^2+3n)/(n^2+3n+2)<1
1/1*2*3+1/2*3*4+……1/n(n+1)(n+2) <0.25=1/4
版权及免责声明
1、欢迎转载本网原创文章，转载敬请注明出处：侨谊留学（www.goesnet.org）；
2、本网转载媒体稿件旨在传播更多有益信息，并不代表同意该观点，本网不承担稿件侵权行为的连带责任；
3、在本网博客/论坛发表言论者，文责自负。
