Let $z$ denote a complex number and $\{\alpha_n\}$ be a sequence in $\ell^2$. Would you help me to prove that series $\sum_{n=0}^{\infty} \alpha_n z^n$ has radius of convergence greater than or equal to $1$. I have proved special cases when $\alpha_n \neq 0$ by the ratio test, but can't do the same for the general one.
Thanks.