Borel summation

Summary

Borel summation is a method that assigns a value to many divergent series, in particular to series whose coefficients grow like factorials. It divides the $k$-th coefficient by $k!$, sums the resulting convergent series, and recovers the lost factorials with a Laplace transform. The method gives the usual sum for every convergent series. It is a basic tool for the asymptotic series of perturbation theory and of differential equations.