Cris Moore [WDS editing a bit & changing to Elkies notation]: Define S(x) = x-x^2+x^4-x^8+... As x->1, S(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 S(x) = x-S(x^2) = x-x^2+S(x^4) = x-x^2+x^4-S(x^8) = etc 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? --That Elkies page DA pointed out http://www.math.harvard.edu/~elkies/Misc/sol8.html says the Fourier series is known in closed form, which should solve your problem and then some. Except he does not say what it is. S(x) is ultimately periodic as a function of y=log_4 ( log(1/x) ) with period=1 and amplitude (says Elkies) approx 0.00275. I'd be interested to know what this fourier series is & how to get it. Let T(y)=S(x) Then x=exp(-4^y) and T(y) = exp(-4^y) - T(y + 1/2) and we are interested in y-->+infinity and in what is the fourier series of T(y).