Free energy

I have a computer in a room that, in winter at least, is quite cold. The computer uses electricity, which runs the computer - generating information - and ultimately ends up as heat.

I could have a heater in the room instead, using the same amount of electricity as the computer, but without generating any information, only heat. It feels like I'm getting my information generated for free, because the computer warms the room as effectively as a heater consuming the same power, but it also generates information without consuming any additional power.

This is obviously wrong, but why?