1/(1*3)+1/(3*5)+1/(5*7)+~~~+1/(2006*2007)

1/(1*3)+1/(3*5)+1/(5*7)+~~~+1/(2006*2007)
=1/2*[(1-1/3)+(1/3-1/5)+(1/5-1/7)+....+(1/2006-1/2007)]
=1/2*(1-1/2007)
=1/2*2006/2007
=1003/2007

=(1-1/3+1/3-1/5+1/5-1/7+...+1/2005-1/2007)/2
=(1-1/2007)/2
=1003/2007
题目应该是1/(1*3)+1/(3*5)+1/(5*7)+~~~+1/(2005*2007)

你的题目有问题，最后是1/(2005*2007)
1/n(n+2)=[1/n-1/(n+2)]/2
原式=[(1-1/3)+(1/3-1/5)+……+(1/2005-1/2007)]/2
=(1-1/2007)/2
=1003/2007

1/(1*3)+1/(3*5)+1/(5*7)+...+1/(2006*2007)
=(1/2)*[1-1/3+1/3-1/5+1/5-1/7+...+1/2003-1/2005]+1/(2006*2007)
=(1/2)*(1-1/2005)+1/(2006*2007)
=1002/2005+1/(2006*2007)

感觉错了 应该最好是1/2006*2008

将它分解成：

1/2*<2/(1*3)+2/(3*5)+............2/(2006*2008)>
=1/2(1-1/2008)
=1/4016