So I have to calculate an integral that involves a Fourier series of some function. I would like to get some kind of local control of the function near zero the series is
$$f(x)=\sum_{n=1}^{\infty}\dfrac{1}{2^n}\cos (x 2^n)$$
I see that I can't do a Taylor expansion of the $\cos's$ and interchange sums. Does anyone have any idea what the local behavior of the function is like?
The integral that I want to find out if it is finite or not is
$$\int_{-1}^{1}\dfrac{1}{1-f(x)}dx$$