1/1*2+1/2*3+1/3*4+1/4*5+1/5*6+···+1/99*100
=(1-1/2)+(1/2-1/3)+(1/3-1/4)...+(1/99-1/100)
=1-1/2+1/2-1/3+1/3-1/4+.1/99-1/100
=1-1/100
=99/100