Is it possible to write 1/3 as an infinite sum of the form a_1 (2/3)^1 + a_2 (2/3)^2 + ... where the sequence of nonnegative integer coefficients a_1, a_2, ... is eventually periodic?

No, I figure. Here's why http://www.graysage.com/djr/propp.pdf -- Don Reble djr@nk.ca