We know that if the power series $\sum_{k=0}^\infty a_kz^k, a_k, z \in \mathbb{C}$ has a radius of convergence $R$ then it converges absolutely for $|z| < R$ and diverges for $|z| > R$ but the behaviour of the series for $|z| = R$ is unknown.
My question is: if we know that $\sum_{k=0}^\infty a_kz^k$ converges absolutely for $|z| = c$ for some $c \in \mathbb{R}$. Do we know if its radius of convergence must be strictly bigger than $c$? i.e. $R > c$?
Best Answer
No. Take, for instance, $\sum_{n=1}^\infty\frac{z^n}{n^2}$. Its radius of convergence is $1$, but it converges absolutely when $\lvert z\rvert=1$.