Let $p_k$ be a complex polynomial with $p_k(0)=1$, with no zero in $\{|z| \le k^3\}$ and $deg(p_k) \le k$. Show that $\prod_{i=1}^\infty p_i(z)$ converges.
One can observe that $p_k(z)=1+h_k(z)$ but I dont know how to determine the convergence of $\sum_{i=1}^\infty h_i(z)$.
If the polynomial $p_k$ has zeros $\zeta_1,\dotsc, \zeta_{d_k}$, where $d_k = \deg p_k$, then you can write
$$p_k(z) = \prod_{j = 1}^{d_k}\biggl(1 - \frac{z}{\zeta_j}\biggr).$$
In that form, with the constraints on the zeros of $p_k$, it is easy to show that the product $\prod\limits_{i = 1}^{\infty} p_i(z)$ converges locally uniformly on $\mathbb{C}$.