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Originally Posted by Reincarnate

The problem lies in the sort of assumption that adding up an infinite number of terms equals infinity.

I wouldn't say this assumption is made in the paradox at all. In fact, it's implied (even if you know no math) that the sum of the infinite number of terms is finite. The thing that makes Zeno's Paradox confusing at all is the implication that, to move at all, you have to complete an infinite number of tasks - even though you might think you could complete them in a finite time, there are still too many to ever count.
of course, the Paradox is a load of bollocks anyway, but it does require some thinking/explanation the first time you see it
To me, the issue Zeno's Paradox brings up is actually quite distinct from real-world questions such as "does this ball bounce an infinite number of times".