The 1st Battalion, 36th Artillery was credited with participating in the Tunisia, Sicily, Naples-Foggia and Anzio Campaigns. After being reorganized and redesignated 36th Field Artillery Battalion in March 1944, its campaign credits include Anzio (final phase), Rome-Arno, Southern France, Rhineland, Ardennes-Alsace and Central Europe Campaigns. The battalion appears to have been assigned to II Army Corps in North Africa and IV Army Corps from just prior to the Salerno Landing. The battalion supported the 36th and 45th Infantry Divisions during the Salerno Landing and appears to have supported the 3d Infantry Division during the Anzio Landing, reverting to Corps control two days after. It landed with the IV Corps, supporting the 3d, 36th and 45th Infantry Divisions in the Invasion of Southern France.
The 3d Infantry was an element of the 6th Infantry Division until relieved from assignment and inactivated in June 1941. In February 1942, the 1st Battalion was reactivated in New Foundland and the remainder of the regiment was reactivated in Greenland. The regiment went to England as a regimental combat team in May or June of 1944 and appears to have been assigned to Third Army. It went ashore at Normandy sometime after 25 July 1944 and participated in the Northern France Campaign until 24 September 1944 (its only campaign credit of the war). The regiment apparently performed garrison duty in France for the remainder of the war and moved into Germany sometime after the surrender where it took part in the Occupation (some elements were active, assigned to the Berlin Brigade, as recently as 1989). I can't find any connection between the 3d Infantry and the 36th Field Artillery Battalion before, during or after the war.