(1/2+1/3+------+1/2009)x(1+1/2+------+1/2008)-(1+1/2+------+1/2009)x(1/2+1/3+------+1/2008)

(1/2+1/3+------+1/2009)x(1+1/2+------+1/2008)-(1+1/2+------+1/2009)x(1/2+1/3+------+1/2008)
(1/2+1/3+------+1/2009)x(1+1/2+------+1/2008)-(1+1/2+------+1/2009)x(1/2+1/3+------+1/2008)

(1/2+1/3+------+1/2009)x(1+1/2+------+1/2008)-(1+1/2+------+1/2009)x(1/2+1/3+------+1/2008)
(1/2+1/3+------+1/2009)x(1+1/2+------+1/2008)-(1+1/2+------+1/2009)x(1/2+1/3+------+1/2008)
=(1/2+1/3+------+1/2009+1-1)x(1+1/2+------+1/2008)-(1+1/2+------+1/2009)x(1/2+1/3+------+1/2008+1-1)
=(1+1/2+1/3+------+1/2009)x(1+1/2+------+1/2008) -(1+1/2+------+1/2008)-
[(1+1/2+------+1/2009)x(1+1/2+1/3+------+1/2008)-(1+1/2+------+1/2009)]
=(1+1/2+------+1/2009)-(1+1/2+------+1/2008)=1/2009

设1/2+1/3+------+1/2008=x
（x+1/2009）*(1+x)-(1+x+1/2009)*(x)
=x^2+x+1/2009+1/2009*x-x-1/2009*x-x^2
=1/2009

（1/2+1/3+------+1/2009）比（1+1/2+------+1/2009）少一项1
（1+1/2+------+1/2008）比（1/2+1/3+------+1/2008）少一项1
展开后
(1/2+1/3+------+1/2009)x(1+1/2+------+1/2008)写成
(1/2+1/3+-----+1/2009)+(1/2+1/3+...

全部展开

（1/2+1/3+------+1/2009）比（1+1/2+------+1/2009）少一项1
（1+1/2+------+1/2008）比（1/2+1/3+------+1/2008）少一项1
展开后
(1/2+1/3+------+1/2009)x(1+1/2+------+1/2008)写成
(1/2+1/3+-----+1/2009)+(1/2+1/3+------+1/2009)x(1/2+1/3+-----+1/2008)
（1+1/2+------+1/2008）比（1/2+1/3+------+1/2008）写成
(1/2+1/3+-----+1/2008)+(1/2+1/3+-----+1/2009)x(1/2+1/3+-----+1/2008）
相减得
(1/2+1/3+-----+1/2009)-(1/2+1/3+-----+1/2008)=1/2009

收起