1/(1×2)+1/(2×3)+.+1/(19+20)
=(1/1-1/2)+(1/2-1/3)+.+(1/19-1/20)
=1/1-1/2+1/2-1/3+1/3-.-1/19+1/19-1/20
=1-1/2+1/2-1/3+1/3-.-1/19+1/19-1/20
=1-1/20
=19/20