Take a look at the capacitor. If it is swollen, rusty, puffed up on top, or leaking it is bad. Sometimes you cannot tell by looking at it so you bench test it. Using a suicide cord attach the aligator clips to the capacitor terminals, Place the cap where if it explodes, it's remains won't get on you. Yes virginia, I have had one blow up in my face.
 TIP: (check to see if the terminals on the capacitor are shorted to avoid any minor explosions.) Plug the cord into a 120 volt outlet for 3 seconds. This charges the cap. Tap the cheater cord prongs on an iron surface to discharge it. If it makes a snap or popping sound it is good. If it does not hold a charge it is bad. Another way  is to check it with an analog meter. Set the meter on the most sensitive continuity setting (ohms). Be sure the capacitor is discharged so the meter won't be fried. Touch the leads to the discharged cap. The needle on the meter should hop; reverse the leads and retry; it should hop again. That is an indication of a cap that will hold a charge. If the needle never moves of course that is a sign the cap is bad(open). Also if the needle goes all the way over and stays, that is an indication the cap is shorted and therefore bad. The cap is designed to boost the power to the motor to get it going with multiple windings being powered up at once. As soon as the motor reaches running speed the cap drops out of the circuit and then it only takes half the power for the motor to continue running at optimum speed. This saves energy.
 FYI: The old style shaded pole motors had no capacitor and so they used twice the power to do the same job.

                        Authentic redneck HVAC slang. Love it