Cauchy product
Theorem 8.1 (Cauchy product). If \(f(z) = \sum a_n (z - z_0)^n\) and \(g(z) = \sum b_n (z - z_0)^n\) both converge on \(\abs {z - z_0} < r\), then on the same disc
\[ f(z) g(z) = \sum _{n=0}^{\infty } c_n (z - z_0)^n, \qquad c_n = \sum _{k=0}^{n} a_k b_{n-k}. \]
Source: [1, ch. 5]