I've never worked on the V8, but mine has the same overflow tank. Oftentimes as the car gets older this needs to be reaplced as the main connection to the radiator (the one underneath the tank) cracks (usually due to someone over-tightening the connection. Of course a dealer would never do such a thing (well it might be less likely) but many mechanics seem to over-tighten the hose clamps that hold down the rubber coolant hoses over plastic necks. These I imagine are people that learned how to fix radiators on cars with metal components- where one could tighten it up so much the hose rubber would be compounded to death. In the case of most BMWs, the necks crack either during tightening or a subsequent heating/cooling cycle. A brand new tank costs somehting like $60 and is easy to install. Be extra careful not to break the brittle 'pisser' line that feeds from just udner the cap, the other hose there is the overflow line which leaks excess coolant out to under the fender on the passenger side. The cap might be faulty (there was a recall on these I think even on your car). But it might be a coolant hose also, there is one on the back of the motor which I think is notorious and dificult. If the car is new to you I would be changing all the coolant hoses and making sure the connections are smooth, clean, not over-tightened and properly sealed. Do a coolant flush at the same time (make absolutely sure you use Original BMW coolant). It will cost a few hundred bucks to do it all (doing the water pump and thermostat is sensible also) but on an old BMW believe me you want this kind of insurance, it will save you from overheating which can cost thousands as the head gaskets are specilaised jobs that often require much labour and other parts.
All the hoses are wear items anyway. Doing all this is one go would give you reliability with which you can enjoy one of the best cars ever put on the road Nick