Given a power series $\sum_{n=0}^\infty a_n x^n \ ,x \in \mathbb R\ $ with radius of convergence $R$.
Is that true that:
If the series does not converge at one boundary, $R$ or $-R$, then the convergence on $(R,-R)$ is not uniform.
If the series converge at $R$ (or $-R$), then the convergence is uniform on $[0,R]$ (or $[-R,0]$). (not just from $0$, from any $r<R$.)