1/1*3*5=1/15 =1/4*(1/1*3-1/3*5),
1/3*5*7=1/105 =1/4*(1/3*5-1/5*7),
.
1/2001*2003*2005=.=1/4*(1/2001*2003-1/2003*2005),
∴原式=1/4*(1/1*3-1/3*5)+1/4*(1/3*5-1/5*7)+.+1/4*(1/1*3-1/3*5)
=1/4*(1/1*3-1/3*5+1/3*5-1/5*7+.+1/2001*2003-1/2003*2005)
=1/4*(1/1*3-1/2003*2005)
=1004003/12048045
若第一项不改,可从第二项开始计算,最后再加第一项,结果为1/6+1/3*5-1/2003*2005