二分之一+六分之一+十二分之一+二十分之一+三十分之一+四十二分之一+五十六分之一+七十二分之一+九十分之这题的规律是什么?还要把算式一步一步算出来,能简略的就简略,知道的麻烦说下

1/(1*2)+1/(2*3)+1/(3*4)+……+1/(9*10)

1/2+1/6+1/12+1/20+1/30+1/42+……+1/90
1/2=1-1/2
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
……
1/90=1/9-1/10
所以原式=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-……+1/9-1/10
=1-1/10
=9/10

裂项求和
1/2+1/6+1/12+1/20+1/30+1/42+1/56+1/72+1/90
=1/1*2+1/2*3+1/3*4+1/4*5+1/5*6+1/6*7+1/7*8+1/8*9+1/9*10
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+(1/6-1/7)+(1/7-1/8)+(1/8-1/9)+(1/9-1/10)
=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8+1/8-1/9+1/9-1/10
=1-1/10
=9/10