求级数∑n=1∞(2n+1)/n2(n+1)2的和．
因为(2n+1)/[n2(n+1)2]=(n2+2n+1-n2)/[n2(n+1)2] =1/n2-1/(n+1)2，所以s=limn→∞sn =limn→∞[(1-1/22)+(1/22-1/32)+…+(1/n2- 1/(n+1)2]=limn→∞(1-1/(n+1)2)=1．
求级数∑n=1∞(2n+1)/n2(n+1)2的和．
因为(2n+1)/[n2(n+1)2]=(n2+2n+1-n2)/[n2(n+1)2] =1/n2-1/(n+1)2，所以s=limn→∞sn =limn→∞[(1-1/22)+(1/22-1/32)+…+(1/n2- 1/(n+1)2]=limn→∞(1-1/(n+1)2)=1．
