Change the speed of an induction motor with an additional capacitor
I had the same task and just connected capacitor in series with fan. It works. Case shifting capacitor was not changed.  Speed of asynchronous induction motors depend of voltage frequency, but rotor turns slower than magnetic field. That turns difference create torgue. Connecting capacitor in series reducing  actual voltage on motor. So for the same load turns difference should increase and motor turns slower.  To achieve desired speed you need to try different capacitor values.