1/3+1/6+1/12+1/24+1/48+1/96+1/192+1/384=?

=1/3(1+1/2+1/4+1/8+1/16+1/32+1/64+1/128)
=1/3(1-1/256)×2
=85/128

1/3+1/6+1/12+1/24+1/48+1/96+1/192+1/384
=1/3+1/(2×3)+1/(4×3)+1/(8×3)+1/(16×3)+1/(32×3)+1/(64×3)+1/(128×3)
=(1/3)×(1+1/2+1/4+1/8+1/16+1/32+1/64+1/128)
=(1/3)×[1+1/2 +(1/2)²+(1/2)³+(1/2)⁴+(1/2)^5+(1/2)^6]
=(1/3)×1×[1-(1/2)^7]/(1-1/2)
=(1/3)×(255/256)/(1/2)
=85/128

2/3-(1/3)*(1/2)^127

1/3(1-(1/2)8)/(1-1/2)
=255/256*2/3
=85/128