= Solution
{c}
If $z_0=0$, there is no $z$ with $|z|<|z_0|$, so the claim is vacuous. Suppose $z_0\ne0$. Convergence of $\sum a_nz_0^n$ implies $a_nz_0^n\to0$, because <terms of a convergent series tend to zero>. In particular this is a <bounded sequence>, so $|a_nz_0^n|\leq C$ for some finite $C$ and every $n$.
For $|z|<|z_0|$, put $q=|z|/|z_0|<1$. Then
$$
|a_nz^n|=|a_nz_0^n|q^n\leq Cq^n.
$$
The <geometric series> and <comparison test for series> prove
$$
\boxed{\sum_{n=0}^{\infty}|a_nz^n|<\infty.}
$$
Thus convergence at one point gives <absolute convergence> at every point strictly closer to the centre.
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