You do realize we have an unlimited energy source burning down on us during the day? Energy isn't the issue here. We already overproduce during the day (though electric cars might take the load off of that in the near future).

Also you don't have to make the carbon go poof, you just have to capture it and store in in another form (that's not in the atmosphere). So yes, if you have a way to capture it and you use solar power (or any other renewable) you can reduce the carbon in the air.

Earth isn't a closed system.