Boundary Values & Fourier Series · Superchallenge problem

The square wave equal to $1$ on $(0, \pi)$ and $-1$ on $(-\pi, 0)$ has partial Fourier sums $S_N(x) = \dfrac4\pi\displaystyle\sum_{k=1}^{N}\dfrac{\sin(2k-1)x}{2k-1}$. The largest value of $S_N$ near the jump at $x = 0$ does not tend to $1$. Find $\displaystyle\lim_{N\to\infty}\max_{0 < x < \pi/2}S_N(x)$, rounded to four decimal places.