Air conditioner inside blower fan wont start, replaced the capacitor and it still will not start automatically?

I bought 3 capacitors to have on hand while im off at basic training and AIT with the army, i tried all of them and none of them start the fan motor. However if i switch the thermostat to fan and i use my hand to spin the squirrel fan it slowly comes up to speed and runs until i switch it off. My assumption is that if the fan will start with a push from my hand it should start with the "push" that the capacitor gives it. What else can i check?