(1/2 3^0 + 1/8 3^2 + ...) / (1/2 3^0 + 1/4 3^1 + 1/8 * 3^2 + ...)
... which can be transformed into an infinite series with a Cesàro sum of 0.5, so that's my answer.
You are taking a Cesàro sum by treating each possible series of coinflips as a summand. This is arbitrary. If you instead took a series over time, it would not have a Cesàro sum. (i.e. for the partial sum at time t, you only count Beauties that might exist in the first t days)
I got into a heated debate a couple days ago with some of my (math grad student) colleagues about the Sleeping Beauty Problem. Out of this discussion came the following thought experiment:
Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: She will be put to sleep. During the experiment, Beauty will be wakened, interviewed, and put back to sleep with an amnesia-inducing anti-aging drug that makes her forget that awakening. A fair coin will be tossed until it comes up heads to determine which experimental procedure to undertake: if the coin takes n flips to come up heads, Beauty will be wakened and interviewed exactly 3^n times. Any time Sleeping Beauty is wakened and interviewed, she is asked, "What is your subjective probability now that the coin was flipped an even number of times?"
I will defer my analysis to the comments.