# Radius of convergence of power series and absolute convergence

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. $$R > c$$?

No. Take, for instance, $$\sum_{n=1}^\infty\frac{z^n}{n^2}$$. Its radius of convergence is $$1$$, but it converges absolutely when $$\lvert z\rvert=1$$.
No the strict inequality need not hold, however, $$R≥ c$$ holds in this case.