You can test the PSU with the following method.
Remove all external devices except for the mouse and keyboard.
Turn of the computer and unplug it from the wall.
Press and hold the power button for 15 to 20 seconds, the power LED should go out.
Open the case and remove the motherboard battery, please note the polarity (+ or -) before removing it.
Press and hold the power button again for ~30 seconds.
Reinstall the battery observing the proper polarity and plug the power cord back into the wall, see if the computer will boot properly when the power button is pressed.