1+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+------+1/(1+2+3+----+100)=

来源：百度知道编辑：UC知道时间：2024/06/21 12:41:55

1+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+------+1/(1+2+3+----+100)
=1+2(1/2*3+1/3*4+1/4*5+......+1/100*101)
=1+2(1/2-1/3+1/3-1/4+1/4-1/5+......+1/100-1/101)
=1+2(1/2-1/101)
=1+2*99/202
=1+99/101
=200/101

1+2+……+n=n(n+1)/2
1/(1+2+……+n)=2/[n(n+1)]=2*[1/n-1/(n+1)]

所以1+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+------+1/(1+2+3+----+100)
=2*(1-1/2)+2*(1/2-1/3)+……+2*(1/100-1/101)
中间抵消
=2*(1-1/101)
=200/101

原式=2[(1-1/2)+(1/2-1/3)+(1/3-1/4)+------+(1/100-1/101)]
=200/101

(1/2005-1)(1/2004-1)........(1/3-1)(1/2-1) 1+1/(1+2)+1/(1+2+3)+...+1/(1+2+3+...+100) 1+1/(1+2)+1/(1+2+3)+-------+1/(1+2+3+----+100) 1+1/1+2+1/1+2+3+...+1/1+2+3...+2000 1+1/1+2+1/1+2+3.........+1/1+2+3.....100 1*(1/1+2)*(1/1+2+3)*~~~*(1/1+2+~~~2005)=? (1-1/2)(1-1/3)(1-1/4)(1-1/5).....(1-1/1000) 1+1/2+1+1/3+1+1/4+......+1/100=? (1+1/2)(1+1/2^2)(1+1/2^4)(1+1/2^8) (1-1/2^2)*(1-1/3^2)*(1-1/4^2).......(1-1/100^2)