I am trying to understand Dyson's argument about the divergent nature of the perturbative expansion in QED. Quoting his own words [Source: hep-ph:0508017 (PDF)]
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 $e=0$?