[math-fun] oscillating series

Define f(x) = x-x^2+x^4-x^8+... As x->1, f(x) becomes Grandi's series 1-1+1-1+..., and it oscillates more and more rapidly around 1/2. These oscillations are periodic in log(1-x): one way to see that is to notice that f(x) = x-f(x^2) or roughly f(x) = 1-f(x^2). So squaring x, which (when x is close to 1) increments log(1-x) by log 2, flips f(x) around 1/2. Hardy knew about these oscillations (see http://en.wikipedia.org/wiki/Summation_of_Grandi%27s_series). But my question is: how do we calculate their amplitude? Numerically, f(x) ranges from 1/2+delta to 1/2-delta, where delta = 0.00274 or so. Does anyone know how to obtain this number? - Cris