利用极限存在准则证明lim(n—>无穷)n^2[1/(n^2+1)^2+2/(n^2+2)^2+...+n/(n^2+n

1。n^2[1/(n^2+1)^2+2/(n^2+2)^2+。。。+n/(n^2+n)^2]≥ n^2[1/(n^2+n)^2+2/(n^2+n)^2+。。。+n/(n^2+n)^2]= n^2[1+2+。。。+n]/[(n^2+n)^2]= n^2[n(n+1)/2]/[(n^2+n)^2]= (1/2)[n^4+n^3]/[n^4+2n^3+n^2](1/2)[n^4+n^3]/[n^4+2n^3+n^2]中令n->∞,极限是1/22。n^2[1/(n^2+1)^2+2/(n^2+2)^2+。。。+n/(n^2+n)^2]≤ n^2[1/(n^2+1)^2+2/(n^2+1)^2+。。。+n/(n^2+1)^2]= n^2[1+2+。。。+n]/[(n^2+1)^2]= n^2[n(n+1)/2]/[(n^2+1)^2]= (1/2)[n^4+n^3]/[n^4+2n^3+1](1/2)[n^4+n^3]/[n^4+2n^3+1]中令n->∞,极限是1/2根据夹逼定理（准则）,知道极限存在,并且极限是1/2。