In Mathematics, one will often want to approximate things. There are two main types of series to do this:
After adding up an infinite number of terms we get to exactly the right answer which will be a finite number, like 'three'. After adding up a very large number of terms we get so close to the right answer we don't care any more and we, happy that we're cool, stop adding on. After all, the rest of the terms get even smaller and smaller, so they're hardly important.
There are several practical problems with the above idea of convergent series. Let us imagine that we are not god, but merely a scientist or engineer. We stand in our lab and see our computer. Our computer can't add up an infinite number of terms, so we're pretty damn sure we can't get the 'exact answer'.
But this is okay - we're a scientist, baby, so just chill. Close enough is close enough, as they say in Reno. We'll just take our convergent series and add up a very large number of terms, right?
We don't have time to add up a very large number of terms. We're busy, damnit. The airflow past this bridge has to be modelled by three o'clock today if we're going to meet our supervisor with anything like a shred of pride, and the finite-time singularity simulation had better be done by Friday or the smartarse down the hall will decide to write the paper without me. Again.
What we want is a very small number of terms we can add up that will give us an alright answer. We don't mind if it's within epsilon of the correct answer - hell, within 20% would be okay - it's a new day and the coffee is strong, but deadlines are deadlines.
This is the idea behind an asymptotic series. We add up a very small number of terms (8, for example, would be a very large number to which to go - 2 is far more likely) which gives almost the exact answer straight away.
Note we don't care what the hell the tail end of the series is doing. If I'm going to add up the first nine terms, I really don't give a damn if the tenth term is 100, 256, or a beautiful rendering of Roger Rabbit in oil on canvas. Most Asymptotic Series are divergent, for a usual definition of divergent.