Power light comes on, but the screen is black, no

The most likely the TV has a power supply related problem. The most common thing to go out is the circuit board that supply power to the screen. Commonly called a Backlight Inverter Board.
I can't say it's 100% the problem because I can't actually look at the tv over the internet.
If by chance the TV is still under warranty, go that route.
If not under warranty and If you have some electronic experience, you can purchase a Service Manual over the internet and diagnose the source of the problem and fix it by soldering in the new part. The Service Manual will give you circuit diagrams, test points, and part# lists. Usually, cost under $20.

I hope I shed some light on the subject for you.