= Solution
{c}
Define the <radius of convergence> of the <power series> by
$$
R=\sup\left\{|w|:w\in\mathbb C,\ \sum_{n=0}^{\infty}a_nw^n\text{ converges}\right\}\in[0,\infty],
$$
with $R=\infty$ when this <set> is unbounded. It is nonempty because the <power series> converges at zero.
If $|z|<R$, the definition of <supremum> supplies a point $w$ of convergence with $|z|<|w|$. The terms $a_nw^n$ of a convergent <series> tend to zero and are bounded, say $|a_nw^n|\leq M$. Therefore
$$
|a_nz^n|\leq M\left(\frac{|z|}{|w|}\right)^n.
$$
The right-hand side has a convergent <geometric series>, since $|z|/|w|<1$. The <comparison test> proves <absolute convergence> at $z$.
If $|z|>R$ and the <power series> converged at $z$, then $|z|$ would belong to the <set> whose <supremum> is $R$, an immediate contradiction. Thus
$$
\boxed{|z|<R\Rightarrow\text{absolute convergence},\qquad |z|>R\Rightarrow\text{divergence}.}
$$
No assertion about the boundary $|z|=R$ follows from this argument. If $R=0$ the first assertion is vacuous; if $R=\infty$ the second is vacuous.
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