Complex Analysis: Theory and Applications · Problem 4.3.2

This is a geometric series in disguise—the key is recognizing that the $n$-th term has the form $a^n$ where $a = \frac{z+i}{2}$. A geometric series $\sum a^n$ converges exactly when $|a| < 1$, which translates here to $\left|\frac{z+i}{2}\right| < 1$, or equivalently $|z+i| < 2$. The disk $D_2(-i)$ is precisely the region where this inequality holds, and outside it the terms don't shrink to zero, so the series must diverge.

This is a geometric series in disguise—the key is recognizing that the $n$-th term has the form $a^n$ where $a = \frac{z+i}{2}$. A geometric series $\sum a^n$ converges exactly when $|a| < 1$, which translates here to $\left|\frac{z+i}{2}\right| < 1$, or equivalently $|z+i| < 2$. The disk $D_2(-i)$ is precisely the region where this inequality holds, and outside it the terms don't shrink to zero, so the series must diverge.