Optimal truncation

Summary

Optimal truncation is the rule of stopping a divergent asymptotic expansion just before its smallest term, for a fixed value of the variable. For many series of factorial growth the error at this point is about the size of the smallest term. This size is exponentially small in the large parameter. So a divergent series can still give an approximation of very high accuracy. The exponentially small error left after optimal truncation is the starting point of hyperasymptotics and of the theory of resurgent functions.