Disc of convergence

Remarks

The series converges absolutely at each $z \in D$, and uniformly on each closed disc $|z - c| \le r$ with $r < R$. It diverges at each $z$ with $|z - c| > R$. The Cauchy–Hadamard formula gives $1/R = \limsup_{n \to \infty} |a_n|^{1/n}$, with $1/0 = \infty$ and $1/\infty = 0$. On the circle $|z - c| = R$, the series can converge at all points, at some points, or at no point. For example, $\sum z^n$, $\sum z^n / n$ and $\sum z^n / n^2$ all have $R = 1$. On $|z| = 1$, the first converges nowhere, the second everywhere except at $z = 1$, and the third everywhere.