解：因为1/n(n+2)=1/2×(n+2-n)/n(n+2)=1/2×[(n+2)/n(n+2)-n/n(n+2)=1/2×[1/n-1/(n+2)]所以，原式=1/2×(1/2-1/4+1/4-1/6+......-1/2006+1/2006-1/2008)=1/2×(1/2-1/2008)=1/2×1003/2008=1003/4016
