There are two identical glasses, one glass is half-full, and the other is half-empty?

Question

Half of water from first glass was poured into the second.
Then, half of water from second glass was poured into the first.

This sequence of steps was continued endlessly.
How much water is poured on each step after sufficiently long time?

Answer 1

1/3 of a glass.
After a long time, the glasses will have ≈2/3 and ≈1/3 in each, and then switch in the next step.
After n pours, the fuller glass has:
(1 - 1/2) + 1/4 - 1/8 + 1/16 - ... + 1/(-2)^(n+1)
and the emptier glass has:
1/2 - 1/4 + 1/8 - 1/16 + ... - 1/(-2)^(n+1)
The limit of these is 2/3, 1/3

Answer 2

1/4 --> 3/8 --> 5/16 --> 11/32 --> 21/64 --> 43/128 --> 85/256

If you look at the even terms, they approach 1/3 from below. If you look at the odd terms, they approach 1/3 from above.

For example:
(2^(2n + 1) + 1) / 3
------------------------- will approach 1/3 as n --> infinity.
... 2^(2n + 1)

The glasses will approach being 1/3 full and 1/3 empty. :-)

Answer 3

I'm thinking 1/3

Answer 4

1/3 of a glass I think.. that's when they reach equilibrium