1/4+1/12+1/24+1/40+1/84+1/112+1/144+1/180

1/4+1/12+1/24+1/40+1/84+1/112+1/144+1/180
=1/4+(1/4-1/6)+(1/6-1/8)+(1/8-1/10)+(1/12-1/14)+(1/14-1/16)+(1/16-1/18)+(1/18-1/20)
=1/4+1/4-1/6+1/6-1/8+1/8-1/10+1/12-1/14+1/14-1/16+1/16-1/18+1/18-1/20
=(1/4+1/4)-(1/6-1/6)-(1/8-1/8)-(1/10-1/12)-(1/14-1/14)-(1/16-1/16)-(1/18-1/18)-1/20
=1/2-1/60-1/20
=13/30
如果在1/40后再加上1/60,计算时会更简便.

1/4+1/12+1/24+1/40+1/84+1/112+1/144+1/180
=1/4*(1+1/3+1/6+1/10+1/21+1/28+1/36+1/45)
=1/4*(1+1/3*(1+1/2+1/7)+1/5(1/2+1/9)+1/4(1/7+1/9))
=1/4*26/15
=13/30

13/30

1/4+1/12+1/24+1/40+1/84+1/112+1/144+1/180
=(1/2+1/6+12/12+1/20+1/30+1/42+1/56+1/72+1/90)/2-1/60
=(1-1/2+1/2-1/3+1/3-1/4……+1/8-1/9+1/9-1/10)/2-1/60
=(1-1/10)/2-1/60=9/20-1/60=26/60=13/30