1/2=1-1/2
1/6=1/2-1/3
1/12=1/3-1/4
.
1/42=1/6-1/7
所以
1/2+1/6+1/12+……+1/42=1-1/2+1/2-1/3+1/3-1/4+……+1/6-1/7=1-1/7=6/7