Defining Convergent and Divergent Infinite Series BC only

A series names three objects and confusing them is the root error of the unit: the terms $a_n$ are a list, the partial sums $S_n = \sum_{k=1}^{n} a_k$ are a second list built from the first, and the sum of the series is the single number $\lim_{n \to \infty} S_n$ when it exists. Convergence is defined only on that middle object, so $\sum a_n = L$ means precisely $S_n \to L$; divergence covers both partial sums that grow without bound and partial sums that oscillate forever without settling, as $1 - 1 + 1 - 1 + \cdots$ does between $1$ and $0$. For $a_n = \left(\frac{1}{2}\right)^{n-1}$ the terms tend to $0$ while the partial sums tend to $2$, two different limits of two different lists.

That $a_n \to 0$ is necessary for convergence and nowhere near sufficient, and the harmonic series $\sum \frac{1}{n}$ is the standing counterexample: its terms tend to zero and its partial sums pass every bound, reaching $10$ only after more than $12{,}000$ terms but never settling. It shares its first two partial sums, $1$ and $\frac{3}{2}$, with the convergent geometric series, and has already passed $2$ by $n = 4$. So a vanishing term can rule a series out and can never rule one in, which is exactly the one-directional statement 10.3 formalises as the nth term test. The two families whose partial sums can be written down, geometric and telescoping, need no test at all: $S_n$ is a formula and the definition applies directly.