A capacitor stores electricity for a short period of time so that the fan motor can receive a smooth flow of power. When a capacitor goes bad, it will not be able to store any electricity which means that the fan motor will not be able to receive a smooth flow of power. This is why you will often see Ceiling Fans with bad capacitors have one blade that is slightly slower than all the others.
If a ceiling fan capacitor goes bad, the fan will not work. A capacitor is an electrical component that stores energy in an electric field. When the capacitor goes bad, it will not be able to store energy and the fan will not work.