From Grimmett's Probability and Random Processes:
Let $G_a(s) := \sum_0^\infty a_is^i$ where $a = \{a_i : i \geq 0\}$ is a real sequence.
Uniqueness. If $G_a(s) = G_b(s)$ for $|s| < R'$ where $0 < R' \leq R$ then $a_n = b_n$ for all $n$.
Is it enough to show that $G_a(s) = G_b(s)$ holds for all $s\in [0,R')$ instead of $s\in (-R',R')$?
Here is the context of my question. I want to understand the justification for the swap in:
$$G_{X_1+\cdots+X_N }(s)=\mathbb E\left(\sum_n \mathbf 1_{\{N=n\}} s^{X_1+\cdots+X_N }\right)= \sum_n \mathbb E\left( \mathbf 1_{\{N=n\}} s^{X_1+\cdots+X_N }\right)$$
As far as I can tell, the swap is unproblematic if $s \geq 0$.