令an=1/(n*2+n+1)+2/(n*2+n+2)+...n/(n*2+n+n)则an>(1+2+..+n)/(n^2+n+n)=n(n+1)/[2n(n+2)]=(n+1)/[2(n+2)]an<(1+2+..+n)/(n^2+n)=n(n+1)/[2n(n+1)]=1/2因为 (n+1)/[2(n+2)]左边当n->无穷时，极限为1/2,由夹逼定理，知an的极限为1/2所以原式=1/2
