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20 1. Mathematical preliminaries

estimate for the remainder, there is no necessity to invoke a special property of the
series (which in any event, perhaps, is not available). Here, we have (from (1.41))

for because (where and so

For any given x, this estimate for the remainder is minimised by the choice n = [x],
exactly as we found earlier. The only disadvantage in using this approach, for any
general series, is that we may not know the sign of the remainder, and so we must
content ourselves with the error
Although a study of series, both convergent and divergent, is a very worthwhile
undertaking and, as we have seen, it can produce results relevant to some aspects of
our work, we must move on. We now turn to that most important class of asymptotic
expansions: those that use a parameter as the basis for the expansion.

1.5 ASYMPTOTIC EXPANSIONS WITH A PARAMETER
We now introduce functions, which depend on a parameter and are to be
expanded as Here, x may be either a scalar or a vector (although our early
examples will involve only scalars). In the case of vectors, we might write (in longhand)
note that commas separate the variables, but that a semicolon is used to
separate the parameter. As we shall see, it does not much matter in this work if the func-
tion we (eventually) seek is a solution of an ordinary differential equation (x is a scalar)
or a solution of a partial differential equation (x is a vector): the techniques are essen-
tially the same. The appropriate definition of the asymptotic expansion now follows.

Definition (asymptotic expansion with a parameter 1)

With respect to the asymptotic sequence defined as we write the
asymptotic expansion of as

for x = O(1) and every The requirement that x = O(1) is equivalently that
x is fixed as the limit process is imposed.

Now suppose that f is defined in some domain, D say, which will usually be prescribed
by the nature of the given problem e.g. the region inside a box which contains a gas. It
is at this stage that we pose a fundamental question: does the asymptotic expansion in