1/1+2+1/1+2+3+1/1+2+3+4+...+1/1+2+3+4...+99
=1/[(1+2)*2/2]+1/[(1+3)*3/2]+1/[(1+4)*4/2]+.+1/[(1+99)*99/2]
=2/(2*3)+2/(3*4)+2/(4*5)+.+2/(99*100)
=2(1/2-1/3)+2(1/3-1/4)+2(1/4-1/5)+.+2(1/99-1/100)
=2(1/2-1/100)
=1-1/50
=49/50