[math-fun] A puzzle I read about

If you haven't seen this one before, it's worth a try: (Provenance to be provided with solution.) << For positive x<1, consider the alternating sum S(x) = x - x2 + x4 - x8 + x16 - x32 + - ... Does S(x) approach a limit as x approaches 1 from below, and if so what is this limit?

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--Dan