1/1*2+1/2*3+1/3*4+1/4*5+.+1/1000*1001

1/1*2+1/2*3+1/3*4+1/4*5+.+1/1000*1001
1/1*2+1/2*3+1/3*4+1/4*5+.+1/1000*1001

1/1*2+1/2*3+1/3*4+1/4*5+.+1/1000*1001
1/(1×2)+1/(2×3)+1/(3×4)+1/(4×5)+……+1/(1000×10001)
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+……+(1/1000-1/1001)
=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+……+1/1000-1/1001
=1-1/1001
=1000/1001

1/1*2+1/2*3+1/3*4+1/4*5+......+1/1000*1001
=1-1/2+1/2-1/3.........................+1/1000-1/1001
=1-1/1001
=1000/1001

由于1/[m(m+1)]=1/m-1/(m+1)
所以原式=（1-1/2)+(1/2-1/3)+(1/3-1/4)+……(1/999-1/1000)+(1/1000-1/1001)
累加 =1-1/1001
=1000/1001