The main reason that France signed a formal aliance with the colonies during the American Revolution was

because they wanted to lower the influence and strength of Britain, which was regarded as the main opponent of France in this period.

Explanation:

After France was defeated in a war with Britain (French and Indian war) they lost all the territories in North America. Additionally, they were practically embarrassed with that treaty and were just waiting for the opportunity to revenge on them. That opportunity finally came in the American Revolution after the battle of Saratoga. At the end peace treaty was once more signed in Paris, this time France was on the side of winners.

an expansion is one of two basic business cycle phases; the other is contraction. the transition from expansion to contraction is a peak, and the changeover from contraction to expansion is a trough.