Are you doing something like this: * Define the series over the 2-adic numbers rather than the reals * Associate every 2-adic integer with its corresponding s

Are you doing something like this: * Define the series over the 2-adic numbers rather than the reals * Associate every 2-adic integer with its corresponding sequence in the inverse limit Z/2^nZ for n = 1..infinity * Truncate the sequence to a certain precision(an index i), and make an equivalence class a ~ b if a and b agree on the first i elements. You get a field Z_(2^n), and identities in 2-adic analysis should carry down to the equivalence classes. Here 1 + 2 + 4 ... = -1 in the 2-adics. That isn't the way I would've thought about signed integers, but it seems like it would work.