Is a power series pointwise convergent in its interval of convergence?

Question

I know a power series is not necessarily uniformly convergent in its interval of convergence, but I was wondering whether it may be pointwise convergent in this interval? If this is false, I would be grateful if a counter example could be provided.

Answer 1

Let $R $ be the radius of convergence of the power series $\sum a_nx^n $.

The convergence is uniform at any compact $[a,b]\subset ]-R,R [$

BUT

if $\sum a_nR^n $ and $\sum a_n (-R)^n $ are convergent then the convergence is uniform at $[-R,R] $.