Why is an infinite series not considered an infinite sum of terms? According to, for example, this excellent page on beginner calculus, an infinite series is NOT an infinite sum of terms.* I'm not even sure what that's asserting.
I think an infinite series is an infinite number of terms that are added together. I also think you can call it the limit of its partial sum $S_n$ as $n \to \infty$ but I'm not sure if or how those two ideas conflict.
Does the infinite sum of terms 'equal' the limit of a series' partial sum, and if not why not?

*To quote: "We do have to be careful with this however. This implies that an infinite series is just an infinite sum of terms and as we’ll see in the next section this is not really true." I looked at the next section and I still don't get what they're saying.


The issue we need to discuss here is that for some series each of these arrangements of terms can have different values despite the fact that they are using exactly the same terms.

That is known as the Riemann series theorem for semi-convergent series; but if you look at a series as "just an infinite sum of terms", this should not hold (the terms are the same, so the sum should be).
