Borel resummation with rapidly growing coefficients

[lokofer]

Borel "resummation"...

Let be a divergent series:

$$\sum _{n=0}^{\infty} a(n)$$ (1)

then if you "had" that $$f(x)= \sum _{n=0}^{\infty} \frac{a(n)}{n!}x^{n}$$

You could obtain the "sum" of the series (1) as $$S= \int_{0}^{\infty}dte^{-t}f(t)$$ in case the integral converges...

- Yes that's "beatiful" the problem is ..what happens if the coefficients a(n) are complicate?..then how can you obtain the sum of the series?...

- By the way i think that Borel resummation can be applied if $$f(t)=O(e^{Mt})$$ M>0, but what happens if f(t) grows faster than any positive exponential?..

 

[fresh_42]

If $f(x)$ is "complicated", then there will be only a numerical integration possible. And if $f(x)$ grows faster than exponential, then the integral likely does not converge.