Final answer:
France and Spain allied with the Americans after the Battle of Saratoga as a strategic move, following a significant American victory that demonstrated the potential success of the revolutionary cause.
Step-by-step explanation:
France and Spain waited until after the Battle of Saratoga to ally with the Americans because they needed to see a demonstration of American resolve and capacity to win. This battle served as a decisive victory that convinced the French of the Americans' potential. They were not simply waiting for British weakness, nor were they unaware of the American struggle; rather, they sought a tangible sign that their support would not be in vain.
The American army's victory in Saratoga provided this assurance, and thus France recognized American independence and formed a military alliance. This was not merely a request for support post-battle or because they had prior commitments to other conflicts; the Alliance with France was strategic and based on a perceived opportunity to both support the American cause and advance their interests against Britain.