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16 1. Mathematical preliminaries

of this text is towards the introduction of methods which aid the description of the
structure of a solution (in the limit under consideration).
Finally, before we move on, we briefly comment on functions of a complex variable.
(We will present no problems that sit in the complex plane, but it is quite natural to
ask if our definitions of an asymptotic expansion remain unaffected in this situation.)
Given and the limit we are able to construct asymptotic
expansions exactly as described above, but with one important new ingredient. Because
is a point in the complex plane, it is possible to approach i.e. take the limit,
from any direction whatsoever. (For real functions, the limit can only be along the
real line, either or However, in general, the asymptotic correctness
will hold only for certain directions and not for every direction e.g. for
(for some and for other args the asymptotic expansion (with
the same asymptotic sequence, fails because for some n.

1.4 CONVERGENT SERIES VERSUS DIVERGENT SERIES
Suppose that we have a function f (x) and a series

then is a convergent series if as for all x satisfying
(for some R > 0, the radius of convergence). This is a statement of
the familiar property of the type of series that is usually encountered; so we have, for
example, as that

and

One important consequence is that we may approximate a function, which has a
convergent-series representation, to any desired accuracy, by retaining a sufficient num-
ber of terms in the series. For example

where the limit as is 2. With these ideas in mind, we turn to the challenge
of working with divergent series.
In this case, has no limit as for any x (except, perhaps, at the one
value x = a, which alone is not useful). Usually diverges—the situation that is
typical of asymptotic expansions—but it may remain finite and oscillate. In either case,
this suggests that any attempt to use a divergent series as the basis for numerical estimates
is doomed to failure; this is not true. A divergent series can be used to estimate f (x)