There are many modern books, for example,
MR1317343 Balser, Werner From divergent power series to analytic functions. Theory and application of multisummable power series. Lecture Notes in Mathematics, 1582. Springer-Verlag, Berlin, 1994.
MR1250603 Candelpergher, B.; Nosmas, J.-C.; Pham, F. Approche de la résurgence. Actualités Mathématiques. Hermann, Paris, 1993.
The very basic idea is the following: You frequently obtain divergent series,
a) as formal solutions of differential (or functional) equations,
b) as perturbation series when you vary a linear operator.
The question is whether these series have any relation to actual solutions of the problem.
It often turns out that they are asymptotic series, and moreover, that they are "Borel summable".
Borel summation is a procedure using a form of Laplace transform that under certain conditions
recovers the function from its formal asymptotic series.