usually fan having two types of windings,i.e starting wdg &
whenever we switch on the supply the fan may take the
moment slowly due to starting wdg,at this time fan requires
some reactive power(because fan wdg acts as inductive
load),at this time capacitor supplies this reactive
power.then only the fan will rotates with its own RPM.
Ac supply given to a equipment having winding then there is a lagging power factor then we use a capacitor which has leading power factor then voltage and current are in phase into some extent which increase power factor.