= Solution
{c}
The <comparison test for series> states that, for $0\leq u_n\leq v_n$ eventually, convergence of $\sum v_n$ implies convergence of $\sum u_n$. Equivalently, divergence of $\sum u_n$ implies divergence of $\sum v_n$.
For $|z|<1$, the coefficient bound gives
$$
\sum_{n=1}^{\infty}|a_nz^n|\leq M\sum_{n=1}^{\infty}|z|^n<\infty
$$
by the <geometric series> formula and the <comparison test for series>. The single term $a_0$ does not affect convergence. For $|z|\geq1$,
$$
|a_nz^n|>m|z|^n\geq m>0\qquad(n\geq1),
$$
so the terms do not tend to zero. The fact that <terms of a convergent series tend to zero> excludes convergence there. Thus
$$
\boxed{R=1.}
$$
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