1/1*2+1/2*3+1/3*4+1/4*5+.........+1/2007*2008+1/2008*2009=

1/1*2+1/2*3+1/3*4+1/4*5+.........+1/2007*2008+1/2008*2009
=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+...+1/2008-1/2009
=1-1/2009
=2008/2009

拆项法啊

1/1*2=1/1-1/2
1/2*3=1/2-1/3
……
类推啊
答案是1-1/2009=2008/2009吧
要是错了自己算一算哈
就是拆项法

1-1/2+1/2-1/3.。。。。。。+1/2008-1/2009=1-1/2009=2008/2009

=1-1/2+1/2-1/3+...+1/2008-1/2009=2008/2009
裂项相消