Suppose $f(z)=\sum\limits_{n=0}^{\infty}a_n z^n$ for $|z|<r$, where $r>0$. Also $f(z)$ is continuous on $|z|\leq r$. My question is whether the function $f(z)$ is analytic in some "big" disk $|z|<R(R>r)$? I think the answer is "yes", because there is no singularity on $|z|=r$, so it can be analytic in a big disk! But I can not give strict proof. Or some theorem which I do not know can imply my question! Any help or hint will welcome!
2026-03-25 02:59:26.1774407566
Power series of analytic function
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Take $\displaystyle f(z)=\sum_{n=1}^\infty\frac{z^n}{n^2}$. Then $f$ is continuous in the region $\{z\in\mathbb{C}\,|\,|z|\leqslant1\}$. And analytic on $D(0,1)$. However, you cannot extend it to an analytic function on a larger circle.