uniform convergence of series at endpoints

Question

I am studying the convergence of a power series, when I encounter this theorem:

Let $\sum a_nx^n$ be power series with a convergence radius of $R$, then for all $0<r<R$ the series converges uniformly on $[-r,r]$. Moreover, if the series converges at $x=R$ then it converges uniformly on $[0,R]$

My question is, if a power series converges within a radius $R$ and moreover, it converges at $x=-R$ and $x=R$, does it follow that it converges uniformly on $[-R,R]$ (closed interval).

It seems to me that we can take $max(N_1, N_2)$ where $N_i$ are the functions of epsilon, but I am not sure if it's true.

EDIT: Let $\epsilon > 0$, then exists $N_1, N_2$ such that for all $n>N_1$, $|f_n(x)-f(x)| < \epsilon$ in $[0,R]$, and for all $n>N_2$ $|f_n(x)-f(x)| < \epsilon$ in $[-R,0]$.

Take $N=max\{N_1,N_2\}$

Is it true?

Answer 1

The answer is yes, iff the series converges at end points.