Is Borel Resummation Always Useful?

[Karlisbad]

"Borel resummation" is useful?

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:

$$a(0)+a(1)+a(2)+..... =S$$

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

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 $$a(n)=(-1)^{n}$$ or $$a(n)=n!$$ 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

 

[fresh_42]

You can ask this about for mathematical "tool", even for Pythagoras. What counts as useful? The Borel measure is part of analysis and analysis is used whenever you have functions. It always depends on how far back you want to trace. Decoherence of quantum mechanics is necessary that you can have a toast, but would you say this? However without, our universe wouldn't exist. Will say: we use the theorems that require Borel measurable sets.