That's basically what plants do. So the question becomes: what is the advantage of your system over biofuels?
An obvious one is that you can use electricity from any source (I am thinking nuclear), while plants are solar-only. But since you mentioned solar electricity, I think a side by side comparison of a biofuel producing field and its factory with a solar farm tied to your system could be interesting.
EDIT: In case it isn't clear, my point is not to call you stupid, I assume everyone who wants to extract CO2 from the air also considered plants, and for some reason, found that artificial means are somehow better. The real question is why? For example I know that natural photosynthesis is not very efficient, but there may be other reasons.