Tried yesterday to do the carrot trick, but I couldnt get the fan to come on at all - never paid any attention before, but does the fan come on all the time, or only when its needed?
As others mentioned, a thermo-viscous fan tends to be on all the time, while an electric one will turn on/off depending on the temperature.
The idea behind a viscous coupling is that you have a silicone-like substance between the fan and the driveshaft for the fan. As the temperature increases, this (viscous) material hardens, and your fan starts turning faster. Depending on the fan/material, it can slip quite a lot when cold, and appear not to turn. (Not to be confused with an electric fan not being on yet).
My advice is to turn on the a/c if you have it, go for a nice ride in heavy traffic. Once your temperature is around 100 (but before it gets into the red- which was 120 if I remember well, don't want to damage the engine!) park the car, and with the engine running open the bonnet. Check your fan.
If it is not turning, you've just found a problem. If you have a viscous fan, the coupling is most probably dead. If you have an electric fan, and it is not on it can be a number of things, starting from the fan itself (stuck?), to the fan motor, wires (corroded? cut?). To be frank, I'm not too clued up on the electric fan model as I have a viscous and all my research is based on the viscous one...
Michele