quantum electrodynamics - Doubt in Dyson's argument about the divergent nature of the perturbative expansion in QED

I am trying to understand Dyson's argument about the divergent nature of the perturbative expansion in QED. Quoting his own words

[...] let

$$F(e^2)=a_0+a_1e^2+a_2e^4+\ldots$$ be a physical quantity which is calculated as a formal power series in $e^2$ by integrating the equations of motion of the theory over a finite or infinite time. Suppose, if possible, that the series... converges for some positive value of $e^2$; this implies that $F(e^2)$ is an analytic function of $e$ at $e=0$. Then for sufficiently small value of $e$, $F(−e^2)$ will also be a well-behaved analytic function with a convergent power series expansion.

My question is, why does the convergence of the series for some positive value of $e^2$ imply that it must be analytic at $e=0$?