Is the fast valid for one who intended to fast a specific day a week in advance, but forgot to renew the intention on the day preceding the fast?

The intention considered valid for obligatory fasting must be made during the night. An intention made before nightfall is not considered. The questioner's fast for the aforementioned day is not sufficient if it relied solely on the intention made a week in advance. Intention is merely the resolve to fast tomorrow, made at any part of the night. So, whenever the thought crosses the heart of the fasting person during the night that tomorrow is Ramadan (definition) and they will be fasting, they have made their intention.