If you have a spare working 24/4 pin psu, use it to test system POST.  if that works it would prove the current psu is failing.
I have read previous posts but am seeking a bit more advice, please. Computer was unplugged for about 5 weeks. Now the power button does not turn on the computer. When plugged in, the green LED light on the back is very faint, barely visible but it is on. The amber LED is illuminated on the mother board. When i removed the power cord, the faint barely visible green led does eventually go out. Having read many posts, I did replace the "button" battery on the mother board. The battery was almost dead when tested. That made no difference. There is no self test button on the back of the unit or on the side of the PSU. Is the fact that the green LED is so faint indicative of something? Like I said, it is difficult to see without shading it but is on.  Thank you for your assistance in advance.   
Studio XPS 8000, no power