How can I either rule out or confirm that my power supply is dead?

Last night I had to reset my breakers for an unrelated reason, so I shut down both of my computers. I got up this morning and tried to boot up my oldest computer, only to find that the computer will not boot up.

When I would press the power button, the power would come on for maybe 1 second, then immediately turn off. I started looking through troubleshooting guides and other members posts here and they have helped a lot. One thing I have noticed, is that most people need unplug their PSU from the wall, or at least flip the switch off then back on in order to get the PC to boot for a second or too again. Mine does not need to be unplug or turned off, I can simply press the power button over and over and it will start to boot for about a second over and over.

I shorted the green and black wire on the large (24 pin) connector and turned the PSU on. It seemed to run just fine (of course, with no real load). I plugged it back in, and switched the PSU to 220v just out of curiosity. It started up, and stayed on this time, but there was no "beep", and the orange indicator light on the front of my tower stayed lit (it has a symbol that looks like a trash bin or cylinder or something). Also, I noticed the CPU fan starts up and runs for about 10 seconds before stopping. Everything else continues to be "running" but I am not getting a video signal to my monitor.

I flipped the PSU back to 110v and tried it again. This time it starts to boot up normally, and stays running for about 5 seconds, but there is no beep, and then it shuts off again. It seems that all I have managed to do, is cause the PC to boot for 5 seconds or so instead of 1.

I tried pulling out the button battery on the motherboard and trying again, but I get the same result. I read I should leave it out for at least 1 hour, so I am trying that right now. Assuming this doesn't work, what would be my next step? Thanks, and sorry I know this was a long read, but I wanted to be thorough.

The PC is a Compaq Presario SR5550F, with an ATI 5770 (I think), and I did have the PSU replaced a few years ago for $50, so the PSU isn't anything to write home about.