Line Segment - Thought Experiment

[Froge]

1 - (1/3) - 2(1/9) - 4(1/27) - 8(1/81)....
= 1 - (2^0)(1/3) - (2^1)(1/3^2) - (2^2)(1/3^3) - (2^3)(1/3^4) - ....
= 1 - Sigma (n=0 -> inf) 2^n / (3^n+1)
= 1 - Sigma (n=0 -> inf) (1/3) (2/3)^n

Consider just the Sigma for a moment. The formula for summation of infinite geometric series is Sum = u1 / (1 - r) where u1 is the initial term and r is the ratio between each terms.

In this series, u1 = 1/3 and r = 2/3, so Sum = (1/3) / (1 - 2/3) = (1/3) / (1/3) = 1

Substitution back into the equation, the the answer is 1 - 1 = 0

None of the segment is left.