what does a blank screen mean for the life of the mac

Hard to say.

Under a year old, tap the power button and see if it wakes up, you may be able to reset the PRAM, reset the SMC, boot into target disk mode, to see if the internal display is truly misbehaving, or it is something more like the graphics card. Of course detach any external displays. If none of the above works, attach an external display with the proper adapter and try to startup the machine to see if it is the graphics chip. If an external display works but the internal display does not, it likely is the graphics chip. Call AppleCare as it is under hardware warranty. if you purchased AppleCare, you have up to 3 years from date of computer's purchase to call AppleCare.

Older than 4 years old, PRAM is no longer an option.

Use the zoom feature of your phone's camera to zoom on the area where the serial number is, and don't post it here, but search on http://checkcoverage.apple.com/ to determine your machine's age.