= Solution
Let $h=f-g$, whose <power series> coefficients are $c_n=a_n-b_n$. Suppose some coefficient is nonzero, and let $N$ be the least such index. Inside the common convergence disc,
$$
h(z)=(z-z_0)^N\left(c_N+\sum_{j=1}^{\infty}c_{N+j}(z-z_0)^j\right).
$$
The parenthesized <analytic function> tends to $c_N\ne0$ as $z\to z_0$, so it does not vanish in some neighborhood. Hence $h$ has no <zeros> sufficiently near $z_0$ except possibly $z_0$ itself. This contradicts $h(z_k)=0$ with $z_k\ne z_0$ and $z_k\to z_0$. Thus \b[all coefficients agree]:
$$
\boxed{a_n=b_n\quad\text{for every }n\geq0.}
$$
This also proves the local isolation of <zeros> of a nonzero <analytic function>.
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