已知.1/n(n+1)=A/n+B/n+1,试求A,B的值,并利用他计算(1)1/1*2+1/2*3+1/3*4+...+1/99*1001/n(n+1)+1/(n+1)*(n+2)+1/(n+2)*(n+3)+...+1/(n+99)*(n+100)
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![已知.1/n(n+1)=A/n+B/n+1,试求A,B的值,并利用他计算(1)1/1*2+1/2*3+1/3*4+...+1/99*1001/n(n+1)+1/(n+1)*(n+2)+1/(n+2)*(n+3)+...+1/(n+99)*(n+100)](/uploads/image/z/2985043-67-3.jpg?t=%E5%B7%B2%E7%9F%A5.1%2Fn%EF%BC%88n%2B1%EF%BC%89%3DA%2Fn%2BB%2Fn%2B1%2C%E8%AF%95%E6%B1%82A%2CB%E7%9A%84%E5%80%BC%2C%E5%B9%B6%E5%88%A9%E7%94%A8%E4%BB%96%E8%AE%A1%E7%AE%97%EF%BC%881%EF%BC%891%2F1%2A2%2B1%2F2%2A3%2B1%2F3%2A4%2B...%2B1%2F99%2A1001%2Fn%EF%BC%88n%2B1%EF%BC%89%2B1%2F%EF%BC%88n%2B1%EF%BC%89%2A%EF%BC%88n%2B2%EF%BC%89%2B1%2F%EF%BC%88n%2B2%EF%BC%89%2A%EF%BC%88n%2B3%EF%BC%89%2B...%2B1%2F%EF%BC%88n%2B99%EF%BC%89%2A%EF%BC%88n%2B100%EF%BC%89)
已知.1/n(n+1)=A/n+B/n+1,试求A,B的值,并利用他计算(1)1/1*2+1/2*3+1/3*4+...+1/99*1001/n(n+1)+1/(n+1)*(n+2)+1/(n+2)*(n+3)+...+1/(n+99)*(n+100)
已知.1/n(n+1)=A/n+B/n+1,试求A,B的值,并利用他计算(1)1/1*2+1/2*3+1/3*4+...+1/99*100
1/n(n+1)+1/(n+1)*(n+2)+1/(n+2)*(n+3)+...+1/(n+99)*(n+100)
已知.1/n(n+1)=A/n+B/n+1,试求A,B的值,并利用他计算(1)1/1*2+1/2*3+1/3*4+...+1/99*1001/n(n+1)+1/(n+1)*(n+2)+1/(n+2)*(n+3)+...+1/(n+99)*(n+100)
提示:1/n(n+1) = ((n+1)-n)/(n*(n+1)) = 1/n-1/(n+1)
故 1/n(n+1)+1/(n+1)*(n+2)+1/(n+2)*(n+3)+...+1/(n+99)*(n+100)的
每一项拆开消项,等于 (1/n-1/(n+1))+(1/(n+1)-1/(n+2))+...+(1/(n+99)-1/(n+100)) = 1/n - 1/(n+100)
= 100/(n+100)
1/n(n+1)=1/n-1/(n+1) 所以A=1, B=-1
因此,计算算式=1/1-1/2+1/2-1/3+1/3-1/4+......+1/98-1/99+1/99-1/100
=1/1+(-1/2+1/2)+(-1/2+1/3)+......+(-1/99+1/99)-1/100=1/1-1/100=99/100