1/2+1/6+1/12+1/20+.+1/10100
=1/1*2+1/2*3+1/3*4+1/4*5+.+1/100*101
=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+.+1/100-1/101
=1-1/101
=100/101