The function that I have had in mind is the power series for $tan^{-1}(x)$ for $|x|<1$. I have long wondered if why exactly we can evaluate at $x=1$.
Abes's Theorem intuition
So if we know that a function converges uniformly on (-R,R), then there exists an $N$ such that $ n\geq N \implies |f(x)-f_n(x)|<\epsilon$ for all $x\in [0,R)$. When we couple this with the knowledge that, since the series converges for $x=R$, there exists an M such that when $n\geq M$, $|f_n(R)-f(R)|<\epsilon$. So we can take n to be greater than the max of N and M? This makes the series uniformly converge for all x in $[0,R]$ and therefore it must be continuous at $x=R$.
I took it that all we really need is uniform-convergence of a continuous function on the interval of interest to guarantee the series converges to a continuous function, so we can take the $\lim_{x\to\ R}$ and obtain the true value of the fucntion.