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Seeking a Geometric Proof of a Generalized Alternating Series' Convergence

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(...