I don't know that it would have anything to do with power supply. If the battery is that depleted, it shouldn't turn on at all.

The fact that it beeps like that could be an indicator that it's not making it right through POST. What is the actual sequence of beep(s) that you get?
Also, do you know what BIOS you have? If it's a normal Dell BIOS check <a href="http://www.bioscentral.com/postcodes/dellbios.htm">here</a> for what the beeps mean.

EDIT: While I'm on the subject, a series of beeps at POST could mean a power supply issue, but it would not be an indicator of the state of the AC adapter. It would be more likely that there is something wrong with the battery housing and/or connections.