1+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)......1/(1+2+3......+2006)

1:1
2:1+2
3:1+2+3
4:1+2+3+4
很明显，当第n个式子时：
n:1+2+3+4+···+n
=n（n+1)/2
1:1/1
2:1/（1+2)
3:1/（1+2+3)
4:1/ (1+2+3+4)
很明显，当第n个式子时：
n:1/(1+2+3+4+···+n)
=1/[n（n+1)/2]
=2/n(n+1)
则原式=1+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+···+1/(1+2+3+···+2006)
=1+2/2×3+2/3×4+2/4×5+2/5×6+···+2/2006×2007
=1+2（1/2×3+1/3×4+1/4×5+1/5×6+···+1/2006×2007）
=1+2（1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+···+1/2006+1/2006-1/2007)
=1+2(1/2-1/2007)
=1+1-2/2007
=4014/2007-2/2007
=4012/2007

1/(1+2+...+n)=2/(n(n+1))=2*(1/n-1/(n+1))

原式
=2*(1-1/2+1/2-1/3+1/3-1/4+...+1/2006-1/2007)
=2*(1-1/2007)
=4012/2007

原式=1+1/[(1+2)*2/2]+1/[(1+3)*3/2]+...+1/[(1+2006)*2006/2]
=1+2/(2*3)+2/(3*4)+2/(4*5)...+2/(2006*2007)
=1+2(1/2-1/3+1/3-1/4+1/4...+1/2006-1/2007)
=1+2(1/2-1/2007)
=1+2005/2007
=4012/2007

小弟弟
1+2+3+4+……+n=n（n+1)/2
那么对任意一个
1/(1+2+3+...+n)=2/(n+1)n=2(1