1+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+.+1/(1+2+3+.+100)
=2/(1*2)+2/(2*3)+2/(3*4)+……+2/(100*101)
=2[(1-1/2)+(1/2-1/3)+……+(1/100-1/101)
=2*(1-1/101)
=200/101