bad battery or bad alternator ?

[rochow@pixelmagic.com]

Scenario: '84 5000S sits for a number of weeks.
I drive it around the block a few times.
It sits another 3 weeks,
Then it won't start, accompanied by warning chimes getting 
fainter.
Jump-start (that isolation-surrounded post is really a good idea!),
drive for about 30 minutes at town speeds.
Car sits another week.
Attempting to start, I get nothing again, and again the 
waring chime gets fainter with every second and with
every turn of the key to Start.

Knee-jerk reaction: must be a bad battery.
Any (dis)agreement based on these symptoms?
(I was about to try and find a new battery, when it occurred
to me that somehow it might be a bad alternator. But the
symptoms don't really fit, do they?)

(BTW: Thanks to all who helped with the cooling fan mystery,
which I will attack this weekend with a newly acquired Bentley,
and also for the battery and Bentley sources)

TIA
-G. Benedikt Rochow