A few more pictures, with greater resolution, fundamental 100 Hz, then 300 Hz, 500 Hz,... 1900 Hz, weighted by 1/n again to create the first ten terms of a square wave. This makes it easier to see how the summed wave shape changes. I used Python for these plots and generated 65,536 samples at 1.024 MS/s. I only plotted one cycle of the 100 Hz fundamental to make it easier to see. There is nothing really new to see, just prettier pictures.
No phase change, group delay is constant, showing the ten signals and their sum is shown first. Notice the "clean" zero crossings at the beginning, middle, and end where all the signals are aligned.
Now leave the first five terms alone (no phase shift), and add increasing phase shift (and thus group delay) to the upper five terms to emulate 1 kHz low-pass filter response (phase only, however, for illustration purposes -- amplitude is not changed). This is similar to what
@pma showed with group delay constant in the midband then changing as the frequencies pass the filter's transition frequency. I also fixed the phase error
@restorer-john noticed so the first five signals align and the next five gradually lag (are delayed) which can be seen looking closely around the zero crossings (no longer "clean"). The difference in the summed signal is again fairly obvious.
HTH - Don