|1/2008-1/2007|+|1/2007-1/2006|+|1/2006-1/2005|+|1/2005-1/2004|+…+|1/3-1/2|+|1/2-1|=

|1/2008-1/2007|+|1/2007-1/2006|+|1/2006-1/2005|+|1/2005-1/2004|+…+|1/3-1/2|+|1/2-1|=
|1/2008-1/2007|+|1/2007-1/2006|+|1/2006-1/2005|+|1/2005-1/2004|+…+|1/3-1/2|+|1/2-1|=

|1/2008-1/2007|+|1/2007-1/2006|+|1/2006-1/2005|+|1/2005-1/2004|+…+|1/3-1/2|+|1/2-1|=
|1/2008-1/2007|+|1/2007-1/2006|+|1/2006-1/2005|+|1/2005-1/2004|+…+|1/3-1/2|+|1/2-1|
=(1/2007-1/2008)+(1/2006-1/2007)+(1/2005-1/2006)+(1/2004-1/2005)+…+(1/2-1/3)+(1-1/2)
=1/2007-1/2008+1/2006-1/2007+1/2005-1/2006+1/2004-1/2005+…+1/2-1/3+1-1/2
=1-1/2008
=2007/2008

可以把绝对值符号去掉，即(1/2007-1/2008)+(1/2006-1/2007)+...+(1/2-1/3)+(1-1/2)观察到前一项的第一个分数和后一项的第二个分数正好消掉，所以答案为1-1/2008=2007/2008