2001 BMW 5 Series; third battery

BMW's will always draw a little bit of current when the cars off. That being said, there is a specification (or tolerance) of how much amperage should be drawn while the car is off. Your E39 should draw less than 40 milli amps after the car goes to "sleep". It takes about 15-17 mins for the car to enter sleep mode. The easiest way you can tell if your car is going to sleep of not is the following method (this can only be said true for the automatic cars)....First, either on a new battery or fully charged battery, start the car and turn on all the accersorys like radio, lights and ect. Turn them all off, then turn the car off. The get out of the car and lock the door. Now what your looking for is the light on the gear selector indicator (little orange light nect to the "p" for park). This light turns off when the car goes into sleep mode. My guess is that your car is not doing this, which basically means your car is NOT going to sleep, thus drawing too much power from the battery while the car is off. At this point, id recommend a BMW tech to diagnois why the car isnt going to sleep. Some of the control modules in the car go bad and casue problems like this. Please let me know what you find out and if this was helpful!!

Check the alternator/connectins/wires to battery to make sure battery is charging. Check the alternator itself if it is giving current to battery. If you are in US it is free at Autozone store
Good Luck