A property of power series (Q1908409)

Using a version of the Rouché theorem the author shows the following result. Suppose that a series \(\sum^\infty_{k = 0} a_k z^k\) is convergent in a disc \(D = \{|z |< R\}\) and let \(\Gamma \subset D\) be a contour with \(0 \in \text{int} \Gamma\). Then for any \(n \geq 0\) there exists \(z_0 \in \Gamma\) such that \[ \left |\sum^\infty_{k = 0} a_k z^k_0 \right |= \left |\sum^n_{k = 0} a_k z_0^k \right |+ \left |\sum^\infty_{k = n + 1} a_k z^k_0 \right |. \]