Final answer:
General Lee invaded the North at Sharpsburg in the fall of 1862 to potentially achieve a military victory for the Confederacy, gain support from European powers, and divert Union troops from other campaigns.
Step-by-step explanation:
General Lee decided to invade the North at Sharpsburg in the fall of 1862 primarily because he believed victory was possible for the Confederacy if they had a successful campaign in Maryland and threatened Washington. Lee also hoped that a significant Confederate victory in the North would convince Great Britain and France to extend support to Jefferson Davis's government and encourage the North to negotiate peace. Additionally, the invasion was intended to divert Union troops engaged in other campaigns, weakening their power over the Mississippi.