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18 1. Mathematical preliminaries

generate a suitable approximation via the familiar technique of integration by parts. In
particular we obtain

and so on, to give

Note that we have used a standard mathematical procedure, which has automatically
generated a sequence of terms—indeed, it has generated an asymptotic sequence,
defined as This is another important observation: our definitions have
implied a selection of the asymptotic sequence, but in practice a particular choice either
appears naturally (as here) or is thrust upon us by virtue of the structure of the problem;
we will write more of this latter point in due course. Here, for the expansion of (1.37)
in the form (1.38), we might regard as the natural asymptotic sequence.
It is clear that we may write, for example,

but what of the convergence, or otherwise, of this series? In order to answer this, we
will use the standard ratio test.
We construct

(because x > 0 and and if this expression is less than unity as for some
x, then the series converges (absolutely). But the expression in (1.39) tends to infinity
as for all finite x; hence the series in (1.38) diverges. To examine this series
in more detail, let us write (1.38) in the form

where the series can be interpreted as an asymptotic expansion for
is the remainder, given by