To be precise, it is not, strictly speaking, calculus which provides the solution to Zeno's paradox. It is the idea of a limit which was missing from the Greeks' mathematical repertoire, and which provides the solution to the paradox. Limits and limiting processes are a very general idea permeating many areas of mathematics, including geometry and metric spaces, topology, and even set theory.

You are, however, quite correct that infinite sums are taught in the calculus syllabus. But even the concept of an infinite collection of objects, which is a very basic idea from something as "simple" as set theory (which actually is highly non-simple) is something that Greek mathematics, which was based in part on the concept of figure and geometry, could not easily cope with. It's all very well to represent a number as a length, and a square number as the area of a square, but an infinite sum of terms is not something which is very easy to draw well.

Hence the problem with Zeno's paradox. One can certainly draw finitely many segments, each of which is half the length of the last. And one can see that the segments do seem to be tending towards a finite length. But since one cannot draw infinitely many segments, one cannot be sure. We can be sure of this with the correct definitions, however, and it was Cauchy who probably provided the best formulation of the limit of a sequence; certainly the whole idea was misunderstood even by great mathematicians before him (and after as well!).