(2n+1)/[n^2(n+1)^2]
=[(n+1)^2-n^2]/[n^2(n+1)^2]
=(n+1)^2/[n^2(n+1)^2]-n^2/[n^2(n+1)^2]
=1/n^2-1/(n+1)^2
所以原式=1/1^2-1/2^2+1/2^2-1/3^2+……+1/n^2-1/(n+1)^2
=1-1/(n+1)^2
=(n^2+2n)/(n+1)^2