5/6=1/2+1/3
7/12=1/3+1/4
9/20=1/4+1/5
11/30=1/5+1/6
.
(18+19)/(18*19)=1/18+1/19
(19+20)/(19*20)=1/19+1/20
所以
5/6-7/12+9/20-11/30+…+〔18+19〕/〔18×19〕-〔19+20〕/〔19×20〕
=[(1/2)+(1/3)]-[(1/3)+(1/4)]+[(1/4)+(1/5)]-[(1/5)+(1/6)]+.+[(1/18)+(1/19)]-[(1/19)+(1/20)]
=(1/2)+(1/3)-(1/3)-(1/4)+(1/4)+(1/5)-(1/5)-(1/6)+.+(1/18)+(1/19)-(1/19)-(1/20)
=(1/2)-(1/20)
=9/20