已知抛物线y=n(n+1)x^2-(2n+1)x+1,在x轴上截得的线段长组成数列an,又它的顶点的纵坐标组成数列bn

题目

已知抛物线y=n(n+1)x^2-(2n+1)x+1,在x轴上截得的线段长组成数列an,又它的顶点的纵坐标组成数列bn
求lim((a1+a2+……+an)-(b1+b2+……+bn))之值

答案

an=|x1-x2|=根号[(x1+x2)^2-4x1x2]=根号{[(2n+1)/n/(n+1)]^2-4/n/(n+1)}=1/[n(n+1)]=1/n-1/(n+1)bn=[4n(n+1)-(2n+1)^2]/[4n(n+1)]=1/4*[1/n-1/(n+1)]a1+...+an=1-1/2+1/2-1/3+...+1/n-1/(n+1)=1-1/(n+1)b1+...+bn=1/...