计算:(1*2+2*3)*(1/1*2+1/2*3)+(2*3+3*4)*(1/2*3+1/3*4)+……+(19*20
来源:学生作业帮 编辑:搜狗做题网作业帮 分类:数学作业 时间:2024/08/19 04:41:41
计算:(1*2+2*3)*(1/1*2+1/2*3)+(2*3+3*4)*(1/2*3+1/3*4)+……+(19*20+20*21)*(1/19*20+1/20*21)=____________
![计算:(1*2+2*3)*(1/1*2+1/2*3)+(2*3+3*4)*(1/2*3+1/3*4)+……+(19*20](/uploads/image/z/18571319-71-9.jpg?t=%E8%AE%A1%E7%AE%97%3A%281%2A2%2B2%2A3%29%2A%281%2F1%2A2%2B1%2F2%2A3%29%2B%282%2A3%2B3%2A4%29%2A%281%2F2%2A3%2B1%2F3%2A4%29%2B%E2%80%A6%E2%80%A6%2B%2819%2A20)
第n项 [n*(n+1)+(n+1)*(n+2)]*[1/n*(n+1)+1/(n+1)*(n+2)]=1+(n+2)/n+n/(n+2)+1=4+2*[1/n-1/(n+2)]
于是,上述和等于
4*19+2*(1/1-1/3+1/2-1/4+1/3-1/6+...+1/18-1/20+1/19-1/21)=76+(1+1/2-1/20-1/21)=76+589/420
于是,上述和等于
4*19+2*(1/1-1/3+1/2-1/4+1/3-1/6+...+1/18-1/20+1/19-1/21)=76+(1+1/2-1/20-1/21)=76+589/420