Radius of convergence greater than 1 if $f$ is analytic in $\left\{|z| < 1 \right\} \cup{} \left\{1 \right\}$ and $a_n \ge 0$

Question

We are doing Laurent series, singularity & Identity theorem at the moment, and I've been struggling to find a lead on this one. $f = \sum_{n=1}^{\infty} a_nz^n$ is analytic in $\left\{|z| < 1 \right\} \cup{} \left\{1 \right\}$ and $a_n \ge 0$, and I am supposed to prove that the convergence radius is greater than 1. I've tried to prove that since I can input 1 into the equation and $f(1)$ is a value, than the sum $\sum_{n=1}^{\infty} a_n$ converges, but that only proves the radius is greater or equal to 1. Feel like I should probably use one of the things I mentioned above, but haven't really been able to find how. Would love even a hint! Thanks :)