Hi, @SimplyBeautifulArt. How do I prove this for the complex case: Let $f(z) = \sum_{n=0}^\infty a_n z^n$ and $g(z) = \sum_{n=0}^\infty b_n z^n$ have the same radius of convergence $R > 0$. Suppose that for some null sequence, $\{z_k\}_{k=1}^\infty \subset B_R(0)$, with $z_k \ne 0$ for any $k$, $\sum_{n=0}^\infty a_n {z_k}^n=\sum_{n=0}^\infty b_n {z_k}^n$. Prove that $a_n = b_n \forall n \in \mathbb {N} \cup \{0\}$.
