Compact convergence $\sum_{n = 1}^{\infty} f(z^n)$

Question

Let $f \colon \mathbb{E} \to \mathbb{C}$ holomorphic with $f(0) = 0$. Ist there a way to prove the compact convergence of the series $$\sum_{n = 1}^{\infty} f(z^n)$$ without using the Taylor series and then applying some theorem for double serieses?