This is a common failure worldwide and BMW carried out a global recall free of charge, I would ask your local main dealership if your car has had this done, if not get them to do it free of charge. If it has been done it will carry a two year parts warranty so should get fixed free of charge.
The A/C has its own radiator that has to be cooled by it's own fan.
BMW honor all recalls on any age of car and as this has been a world wide recall for the 5series E39.
It might need a call from the dealership warranty admin person to BMW customer services dept.
Remember the dealership would prefer to charge retail rates instead of warranty rates so be persistant.
The fans are readly avaiable from aftermarket suppliers and take about 2-3 hours to fit.
Fit as follows
Remove both front lights
Remove front bumber skin
Remove front panel
Remove fan housing
Refit in reverse order
Yes
If the fan does not work the A/C condensor (radiator) overheats, to protect the compressor from damage from overheated gases the ECU stops the A/C from working.
Once you have replaced your fan the system will operate correctly.
BMW Technician
City & Guilds NVQ Level II & III in Vehicle, Mechanical & Electronic Systems BMW Technician Status