I've read that in the Gibbs phenomenon, partial Fourier series will over- or underestimate a function's value in neighborhoods of jump discontinuities. Specifically, the maximum error will converge to the height of the jump discontinuity times $ \mathrm{Si}(\pi)/\pi - 1/2 \approx $ 8.949%. But I notice, visually at least, that there are also 2nd, 3rd, 4th etc. artifacts behind the first that also appear to converge to a fixed proportion of the height in the limit. I've highlighted these artifacts using the approximation for the square wave picture from Wikipedia (link below).
Based on the calculation done at Wikipedia, I surmised these proportions could be explicitly calculated, via
$ \frac{\pi}{4} + (-1)^{n-1} \frac{\pi}{2} e_n = \lim_{N\to\infty}S_N f \left( \frac{2\pi n}{2N} \right) \quad \dots \implies $
$ (-1)^{n-1} e_n = \frac{n}{\pi} \mathrm{Si}\left( \frac{\pi}{n} \right) - \frac{1}{2} . $
However, this can't be correct because it predicts that the error of the artifacts grow larger in the sequence instead of smaller (where the latter is obviously the reality). What gives?