1+1/2+1/1+2 + 1/1+2+3 +1/1+2+3+4 + ....+ 1/1+2+3+..+2009

1+2+ ……+2009=(1+2009)+(2+2008)+……+(1004+1006)+1005=2010*1004+1005
=1005*2*1004+1005
=1005*(2008+1)
=2009*2010/2

1+2+……+2008=(1+2008)+(2+2007)+……+(1004+1005)
=2009*1004
=2008*2009/2

同理
1+2+……+2007=2007*2008/2
……

所以原式=1+1/2+2/2*3+2/3*4+……+2/2009*2010
=1+1/2+2*(1/2-1/3+1/3-1/4+……+1/2009-1/2010)
=1+1/2+2*(1/2-1/2010)
=5023/2010

出来告诉我 题目挺有意思的 我懒得算

=1+1/2+1/[（1+2）×2÷2]+...1/[(1+2009)×2009÷2]
=1+1/2+2/2×3+2/3×4+...+

2/2008×2009+2/2009×2010

=1+1/2+2×（1/2-1/3+1/3-1/4+...+1/2008-1/2009+1/2009-1/2010）
=1+1/2+2×（1/2-1/2010）
=1+1/2+1-1/1005
=2又1003/2010

1+1/2+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+…+1/(1+2+3+…+2009)
=1+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+…+1/(1+2+3+…+2009)+1/2
=1/(1/2*1*2)+1/(1/2*2*3)+1/(1/2*3*4)+1/(1/2*4*5)+…+1/(1/2*2009*2010)+1/2
=2/(1*2)+2/(2*3)+2/(3*4)+2/(4*5)+…+2/(2009*2010)+1/2
=2*[1/(1*2)+1/(2*3)+1/(3*4)+1/(4*5)+…+1/(2009*2010