设1/2+1/3+...+1/2007=a,1/2+1/3+.+1/2006=b
原式=a*(b+1)-(a+1)*b
=(ab+a)-(ab+b)
=a-b
=(1/2+1/3+...1/2007)-(1/2+1/3+.+1/2006)
=1/2007