\[S_{n}(x)=\sum_{k=0}^{n-1}\frac{\sin\left((2k+1)x\right)}{2k+1}=\frac{1}{2}\int%
_{0}^{x}\frac{\sin\left(2nt\right)}{\sin t}\,\mathrm{d}t=\tfrac{1}{2}%
\operatorname{Si}\left(2nx\right)+R_{n}(x),\]