Uniformly convergence on the convergent circle for complex power series!

Question

The power series $f(z)=\sum_{n=0}^{\infty}a_nz^n$ has convergent radius $r>0$, so $f(z)$ is analytic in $|z|<r$. MOREOVER, we suppose $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is continuous on the closed disk $|z|\leq r$. My question is that: Does the series $\sum_{n=0}^{\infty}a_nz^n$ converge uniformly on the closed disk $|z|\leq r$? In my opinion, I can give a example such as: the series $$f(z)=\sum_{n=1}^{\infty}\frac{z^n}{n^2}$$ has radius $r=1$ and converges uniformly on the closed unit disk $|z|\leq1$. I do not know whether my question is right or wrong. Any hints will welcome!