# Borel resummation is useful?

1. Oct 21, 2006

"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