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1.3 ASYMPTOTIC SEQUENCES AND ASYMPTOTIC EXPANSIONS
First we recall example (1.32), which epitomises the idea that we will now generalise.
We already have

and this procedure can be continued, so

(and the correctness of this follows directly from the Maclaurin expansion of sin(3x)).
The result in (1.33), and its continuation, produces progressively better approximations
to sin (3x), in that we may write

and then

At each stage, we perform a ‘varies as’ calculation (as in (1.33), via the definition of‘~’);
in this example we have used the set of gauge functions for n = 0, 1, 2, ....;
such a set is called an asymptotic sequence. In order to proceed, we need to define a
general set of functions which constitute an asymptotic sequence.

Definition (asymptotic sequence)
The set of functions is an asymptotic sequence as
if

for every n.

As examples, we have

(In each case, it is simply a matter of confirming that Some
further examples are given in Q1.9.