How can you test the capacitor used in a ceiling fan?

Assuming you don't have a device for testing capacitors, a somewhat useful method of testing a capacitor is:

Remove the capacitor from circuit.

Using an analog ohm meter (with a needle rather than a digital readout), connect the two leads of the meter to the two leads of the capacitor.

You MAY see the needle jump, but leave the meter connected for a few seconds and see how far the needle falls.

Quickly reverse the polarity of the meter leads.

The needle of the ohm meter should definitely jump then settle back, approaching open circuit.

If you have consistently high resistance (low ohms) the capacitor is faulty, but if the resistance slowly falls (higher ohms), the capacitor is good.

If the capacitor does nothing, consistently high ohms or consistently low ohms, toss the capacitor and get a new one; they're relatively cheap.
touch it with a finger
Lick it. If you die, it works.