作业帮1+1/1+2+1/1+2+3...+1/1+2+3+...+n=
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/12 09:05:11
1+1/1+2+1/1+2+3...+1/1+2+3+...+n=
1+1/1+2+1/1+2+3...+1/1+2+3+...+n=
1+1/1+2+1/1+2+3...+1/1+2+3+...+n=
1+2+3+...+n=n(n+1)/2
1/(1+2+3+...+n)=1/[n(n+1)/2]=2/n(n+1)=2[1/n-1/(n+1)]
于是
原式
=2[1/1-1/(1+1)]+2[1/2-1/(2+1)]+2[1/3-1/(3+1)]+……+2[1/n-1/(n+1)]
=2[1/1-1/2]+2[1/2-1/3]+2[1/3-1/4]+……+2[1/n-1/(n+1)]
=2[1/1-1/2+1/2-1/3+1/3-1/4+……+1/n-1/(n+1)]
=2[1/1-1/(n+1)
=2-2/(n+1)]
=2n/(n+1)