 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,The parenthesized analytic function tends to $c_N\ne 0$ 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 all coefficients agree:This also proves the local isolation of zeros of a nonzero analytic function.