Different bios's have different beep patterns what bios is it? If it's:
If an Award bios, the only BIOS beeps you will hear are 1 Long, and some short combination after. They all indicate a problem either with memory or video, as these are the only problems that would keep the system from doing anything. Any other errors will be given an on-screen error.
1 Long Beep indicates a problem with memory in the first bank, usually an unseated memory module
1 Long, 2 Short indicates a problem with the video card or the memory on it
1 Long, 3 short also indicates a problem with video or the memory on video card
Continuous this usually indicates a problem with the memory, but sometimes is a video problem.
Also dell isn't the best at "fixing" problems. The laptop could have been jolted during tranport and this could be causing your problems! May be a loose mem module!?
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Sounds like it might be a power problem, so could be the battery or battery board. If you've sent it back once though, send it back again and make them fix it properly.