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1/(n2+n)+ 2/(n2+n)+...+n/(n2+n)<1/(n2+1)+ 2/(n2+2)+...+n/(n2+n)<1/(n2+1)+2/(n2+1)+...+n/(n2+1)(1+2+...+n)/(n2+n)<1/(n2+1)+ 2/(n2+2)+...+n/(n2+n)<(1+2+...+n)/(n2+1)[n(n+1)/2]/(n2+n)<1/(n2+1)+ 2/(n2+2)+...+n/(n2+n)<[n(n+1)/2]/(n2+1)?<1/(n2+1)+ 2/(n2+2)+...+n/(n2+n)<(n2+n)/(2n2+2)而lim (n2+n)/(2n2+2)n→∞=lim (1+ 1/n)/(2+ 2/n2)n→∞=(1+0)/(2+0)=1/2由夹逼准则，得：lim 1/(n2+1)+ 2/(n2+2)+...+n/(n2+n)=1/2n→∞