Why A Capacitor Is Used In Series of an Ordinary Fan

to reduce reactance and improve active power part.

Capacitors are used in ceiling fans so that their direction can be changed. The motor is constructed so that starting windings (which actually stay energized in most fans) can be switched to the capacitor circuit, which shifts the phase of the current in one of two sets of start windings, thereby determining which direction the fan starts.

Normally for FHP 1phase motors the capacitor is used to split the voltage part in two vectors to generate starting torque. Shunt capacitor is required in ckt for this purpose. I understand the question is about series capacitor. In some cases a combination of series and shunt capacitors also used.