>>13797504First, let's generalize a little bit. Let's have the zeroth cycle go from to , and let's have evry next cycle have times the wavelength of the previous one. In your case and . Now, let's make the ansatz assuming our function is . Now, what we can do is define the sequence of wavelengths . Of course, we note that the overall function we are looking for must reach zeros at the wavelength partial sums (for example, in your case the wavelength sequences are 1, 0.5, 0.25, ... so it must be zero at 1, 1.5, 1.75, ...). You can easily find an explicit function for the partial sums . Now, here is the interesting part - we can define and then note that is itself a geometric series. Without loss of generality, we make the substitution and then we note we automatically get the behavior we want if and . From here, its easy to see that this can be satisfied with functions of the form . Plugging in the numbers for your specific case, we get .
