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My question is if this is nothing but a "math tool" but not valid for realistic example, for example if we wish to calculate the divergent (but Borel summable) series:

[tex] a(0)+a(1)+a(2)+..................... =S [/tex]

then you take the expression : [tex] B(x)=\sum_{n=0}^{\infty}\frac{a(n). x^{n} }{n!} [/tex] ,

so the sum of the series is just "defined":

tex] a(0)+a(1)+a(2)+.....................=S=\int_{0}^{\infty}dxB(x)e^{-x} [/tex]

Of course if [tex] a(n)=(-1)^{n} [/tex] or [tex] a(n)=n! [/tex] then it's very easy to get B(x), but in a "realistic" situation that you don't even know the general term a(n) or it's very complicated there's no way to obtain its Borel sum

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# Borel resummation is useful?

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