The power to the computer is fine. The green light on the back of my computer comes on then goes off.
Depending upon the PC Manufacturer, this is not indicative that the power is fine, it may merely be saying that the AC Power is on. The fact that you do not hear the power supply fan, integral to all PCs except for the Mac Cube types, makes it likely to be Power Supply death, especially given the suddenness of the problem.
Can you hear the fan running when you turn on the PC? When you say the green light comes on then goes off, is that instantly?
The power supply may be fine, but the power cable may be bad. Or, there may be a loose cable within the box itself. To check for a loose cable, unplug the box from the outlet, then open the box. Be careful as you don't want static to discharge, ruining components within the box. As for a power cable, they're inexpensive and any computer store should have them available. That's the first thing I'd try.
Your computer needs several different levels of power. It’s quite possible to have one or more of those levels just right but another be really off. Power supplies are notoriously fickle.