Continuity of $\sum_{n\geq 0} a_n x^n$ on $(-1,1)$

Question

If the series $\sum_{n\geq 0} a_n$ converges, does it follow that the power series $\sum_{n\geq 0} a_n x^n$ converge to a continuous function on $(-1,1)$?

Will showing uniform convergence of $\sum_{n\geq 0} a_n x^n$ help?

Answer 1

For $|x| \leq 1-\epsilon$ the series $\sum |a_n| |x^{n}|$ is dominated by the convergent series $C(1-\epsilon)^{n}$ where $C=\sup ]\{|a_n|:n\geq 1\}$. By M-test, the given series converges uniformly for $|x| \leq 1-\epsilon$ and hence its sum is continuous there. Since $\epsilon $ is arbitrary we are done. Note that $C <\infty$ because $a_n \to 0$.