1+1/1+2+1/1+2+3+1/1+2+3+4+.1/1+2+3+.+1000
=1+1/[(1+2)×2÷2]+1/[(1+3)×3÷2]+1/[(1+4)×4÷2]+.1/[(1+1000)×1000÷2]
=1+2/(1+2)×2+2/(1+3)×3+2/(1+4)×4+.2/(1+1000)×1000
=1+2×(1/2-1/3+1/3-1/4+1/4-1/5+.+1/1000-1/1001)
=1+2×1/1001
=1又2/1001