 Solution

On $|z|\ge \rho$, put $w=1/z$, so $|w|\le 1/\rho$. The Taylor polynomialsconverge uniformly there to $e^w$. MoreoverFor all sufficiently large $n$, the uniform error is below half this lower bound, so $|f_n(z)|>0$ whenever $|z|\ge \rho$. Every zero therefore lies in $|z|<\rho$.