Doubt in Dyson's argumet about the divergent nature of the perturbative expansion in QED

Question

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)]

... 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 $e=0$?

Comments

Your question in the text is purely mathematical. It concerns all functions.

As to real doubts about Dyson's argument, I have already written many times that the used in practice series are not in powers of $e^2$ or $\alpha$. For example, the meaningful inclusive cross section is a selective sum over all orders of $\alpha$: $F(\alpha)=G(\alpha) + b_0+b_1\cdot\alpha+...=G(1/137) + b_0+b_1\cdot\alpha+...$. First, G(1/137) is finite and the remaining series is IR finite too. Next, all non-analyticity may be contained in $G(\alpha)$ and the remaining series may represent an analytical function. This question has never been studied, so the Dyson's argument is just wrong as dealing with a wrong object - an IR-divergent series.

You can read a toy model about it: https://www.academia.edu/14945605/On_integrating_out_short-distance_physics

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@VladimirKalitvianski: The Dyson series is being used in may places for finding the magnetic moment of the electron at first nontrivial order. Higher order computations of magnetic moments, lamb shifts, etc.  in QED are done in NRQED, which is a joint expansion in $\alpha$ and $Z\alpha$; see. e.g., this article by Kinoshita.

Your method, on the other hand, has never been used in QED, only in your toy models. 

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@ArnoldNeumaier : The question is not about "my method", but about QED series. And "my method" is about possibilities of representation of a function in different ways. Factually, it is already written generally as $F(x)=G(x)$+another series in my comment above, without necessity to look into details of "my method".

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@VladimirKalitvianski: The question is explicitly about the Dyson series in QED, not about resummation techniques that rearrange terms. So strictly speaking your comment was off-topic. 

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@ArnoldNeumaier : "Resummation" is a necessity and banality in QED; it is given in all QED textbooks, but if you think that I am off-topic, you may delete my comments entirely.

Answer 1

Analytic at $x=0$ means convergent for $x$ in the interior of some disk around zero; its maximal radius $r$ is the radius of convergence of the series. The root test for the convergence of an infinite series implies that if a power series in $x$ converges for some nonzero $x$ then the radius of convergence is at least $|x|$. In particular, if the Dyson series converges for some $e^2=x>0$,  it also converges for $e^2=-x/2<0$.