Wylie Coyote
Generally, the capacitor being bad would cause the fan to cease functioning. It is used to give the motor a boost in energy in starting. You test a capacitor using a procedure called a squelch test. Just looking at the capacitor can be a clue, as the capacitor will generally swell when they go bad. Same thing happens in HID lighting.

Flying Dragon
A few simple things to look at before calling someone: Shut off the power and open up the access panels on the air conditioner (be careful, if the capacitior is good, it may hold a charge even with power off, so be very careful where you stick your fingers!) Sometimes the capacitior is good but one of the connecting wires to it (or to the compressor) has burnt off (usually at one of the crimp-on terminals). These wires take a big surge of current for a moment every time it starts and this happens quite often. If the capacitior and wires look OK, also check and see if there is a over temperture/over current sensor, and that it is still good and not tripping out when it tries to start. Some units have a separate relay for the compressor and possibly that is blown out or not turning on for some reason. If you can't find anything obvious, you could try substituting a known good capacitor and see if it starts up. Note: some units have separate capacitors for the fan and compresser, some use one capacitor (which actually has 2 separate capacitors in one can with a common connection and thus, 3 terminals). Quite often one part will be OK and the other blown, so it is possible for the fan to run with the compressor capacitor blown.