18 May
2005
18 May
'05
12:05 p.m.
The simple limit argument is justified because S(x) is monotonically increasing as x --> 1 from below along the real axis. This is clear by grouping the terms of the series as
S(x) = (x - x^2) + (x^4 - x^8) + ... .
Each of those summands is increasing for x near enough 1, but that doesn't imply that the whole sum is, unless I'm missing something important. A bit of numerical experimentation suggests (1) that in fact the limit doesn't exist and (2) that this shouldn't be too hard to prove with a bit of brute force. But I haven't attempted to make good on (2), so (1) might be wrong. :-) -- g