方法一：很简单 1/2=1-1/2
1/6=1/2-1/3
1/12=1/3-1/4
……
1/42=1/6-1/7
相加只剩下1/2 1/2 -1/7
即为6/7
方法二；1/6=1/2-1/3
1/12=1/3-1/4
1/20=1/4-1/5
1/30=1/5-1/6
1/42=1/6-1/7
1/2+1/2-1/7=6/7
这叫裂项求和
方法三：1/2+1/6+1/12+1/20+1/30+1/42
=1-1/2+1/(2*3)+1/(3*4)+1/(4*5)+1/(5*6)+1/(6*7)
=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7
=1-1/7
=6/7
参考资料：若有疑问可以和我交流,我非常欢迎,