1/2+1/6+1/12+...+1/56
=1/2+1/(2*3)+1/(3*4)+……+1/(7*8)
=1/2+(3-2)/(2*3)+……+(8-7)/(8*7)
=1/2+1/2-1/3+1/3-1/4+……+1/7-1/8
=1-1/8
=7/8