Asymptotic series I have found many references to Poincaré and Borel in relation to their work on asymptotic series, but so far, every source I can get my hands on is very old, hence hard to read (this is not true in general, but in this case, texts that predate Oh notation tend not to be clear).

Can you explain the idea behind asymptotic series, give an illuminating example, and/or suggest a good modern exposition of the theory?

 A: Web pages of

Michael Berry: http://www.phy.bris.ac.uk/people/berry_mv/dingle.html

and

John Boyd: http://www-personal.umich.edu/~jpboyd/

both include helpful published and publicly available sources. 
A: 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. 
A: In N. G. de Bruijn's Asymptotic Methods in Analysis (link), Introduction, pp. 11-19, you may find on this topic the following sections, which use the Bachmann-Landau O-notation: 


*

*1.5 Asymptotic series, 

*1.6 Elementary operations on asymptotic series, and 

*1.7 Asymptotics and Numerical Analysis.


This is a 1981 Dover Book republication of the North-Holland Publishing Co. 1970 edition; 1st edition 1958.
The classic treatise is the 1949 book by G. H. Hardy's, Divergent Series, reprinted in 1991 by The AMS Chelsea Publishing, 2nd. edition (link).
