Let $z \in \mathbb{C} \backslash \lbrace 1 \rbrace$ with $|z| = 1$. We consider the following infinite series, which necessarily converges: $$S(z) := \sum_{n = 1}^{\infty}\frac{z^n}{n}$$

Note that $S(-1)$ is the alternating harmonic series.

A straightforward application of the Dirichlet Convergence Test proves any such series converges, but I feel this is a bit like killing a fly with a sledgehammer. (I realize some of you might not think this test is a sledgehammer; I wonder also whether this series is a fly.) In any event, I'm wondering whether there is a way to prove convergence using only a simple geometric argument (with some basic analysis).

For example, we can think of $S(i)$ as taking steps in the plane of length $1/n$, but turning ninety degrees after each one. Then the partial sums correspond to a nested sequence of squares, where the area of the squares is clearly converging to $0$. Thus, an argument using the Nested Interval Property (or really its corresponding $2D$ version) indicates that the series converges.

More generally, I'd think that because we are taking steps of size decreasing to $0$ and rotating by the same amount after each step, there should be a general geometric argument for why $S(z)$ will converge. Ideally, I'd like to have a proof that could be made accessible to early Calculus students, even if not every step is presented in fully rigorous form.

For clarity's sake, I will directly state my **question:** How does one prove $S(z)$ converges using a simple geometric argument that relies at most on basic analysis (e.g., makes no appeal to stronger theorems from Complex Analysis)?