Radius of convergence of a power series convergent for $s \in [0,1)$

Question

Suppose a power series converges to a holomorphic function $F(s)$ for $s \in [0,1)$.

I think that in that case, the power series converges to $F(s)$ for $s$ (complex) such that $|s|<1$, but i don't how to prove it...

Is enough to use rotations? I can't see how to prove it...

Thanks, and if you found it in any book, tell me please.

Answer 1

If a power series centered at $0$ converegs at some $z\in\mathbb C$, then it converges absolutely at every $w\in\mathbb C$ such that $|w|<|z|$.

Since your series converges for every $s\in[0,1)$, it follows from the previous paragraph that it converges absolutely for each $z\in\mathbb C$ such that $|z|<1$. Therefore, the radius of convergence is at least $1$.