If you have tried re-seating the video cables, trying an external monitor, etc, as in the previous posts, and have had no success, then it is still most likely the logic board.
If the capacitors are okay there are plenty of other components on a logic board that can fail. If you can remove the logic board, carefully check the soldering points for deterioration or even for the greenish decay left behind by humidity getting into the machine.
Especially check out the areas where liquid may have gotten into the machine, ie: the ports, the RAM slots and the optical drive slot.