1x1/2+1/2x1/3+1/3x1/4+.+1/19x1/20
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+...+(1/19-1/20)
=1-1/2+1/2-1/3+1/3-1/4+...+1/19-1/20
=1-1/20
=19/20