The sum of the first n terms is the sum from 1 to the nth term, which is the sum of finite terms. Different n terms can get different sums of the first n terms.

The sum of items is the sum of all items (from item 1 to the last item, if there is one). For series, if it is a finite term, it is a number; If it is an infinite term, it is also a number when it converges, and it does not exist when it does not converge (diverge).