Indeed, an asymptotic expansion is NOT a series but a polynomial approximation in a neighborhood of a point so that the function is equal to its polynomial approximation plus a remainder. For instance Taylor's formula with remainder is an asymptotic expansions but it often happens, in number theory for instance, that only asymptotic expansions with one or two terms are known
Edmund Landau's Little oh notation is often used to give the size of the remainder. See Apostol p370
I am not familiar enough with Wikipedia at this time to edit the page but this is a very serious confusion.
See for instance http://fr.wikipedia.org/wiki/Développement_limité