Dan Asimov asked:

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?

The limit does not exist. Since S(x) = x - S(x^2), the limit would have to be 1/2 if it existed. But I'll show that there are values of x arbitrarily close to 1 for which S(x) > 0.5008. Rewrite the definition as S(x) = SUM (x^4^k - x^(2*4^k)) k>=0 For 0 < x < 1, the summands are all positive, so any partial sum is a lower bound for S(x). In particular, calculation shows that, for x=199/200, the sum with k going from 0 to 5 is 0.50088158499... > 0.5008, so S(199/200) > 0.5008. Next note that S(x) = x - x^2 + S(x^4) > S(x^4). So S((199/200)^(1/4)) > S(199/200) > 0.5008, S((199/200)^(1/16)) > S((199/200)^(1/4)) > 0.5008, S((199/200)^(1/64)) > S((199/200)^(1/16)) > 0.5008, etc. Since the arguments of S approach 1, the limit can't be 1/2, so it doesn't exist. Dean Hickerson dean@math.ucdavis.edu