Where are power series uniformly continuous? As far as I know, $f(x)=\sum\limits_{n=0}^\infty a_n(x-x_0)^n$ is continuous on the whole convergence interval $K:=\{x\in\mathbb R:|x-x_0|&lt;r\}$. Is there anything we could say about uniform continuity?

Added: Would it be correct to claim the following? 


  Power series is uniformly continuous on $\bar K:=\{x\in\mathbb R:|x-x_0|&lt;\bar r\}$ where $0&lt;\bar r&lt;r$. 


I'm asking for the clarification since we haven't defined compactness yet.
Check relevant theorems- Weierstrass M-test can be used to prove the uniform continuity.
Check relevant theorems- Weierstrass M-test can be used to prove the uniform continuity.