Influence of Saratoga on French Alliance

During the American Revolution, various military and political strategies were employed by both the Continental Army and the British forces. As the conflict progressed, the turning points of the war became increasingly crucial for the success of the American colonies in their struggle for independence. One such pivotal moment occurred when there was a significant shift in France's attitude towards supporting the American cause. What event is widely considered to have influenced France's decision to formally ally with the American colonies?