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It is convenient, because it simplifies the details, if we elect to work with

and then we have

and so on. Thus, using (1.36a,b), we obtain

so that, in general,

The best estimate, for a given x, is obtained by choosing that n which minimises the
smaller of these two bounds; in this example, this is clearly n = [x] (where [ ] denotes
‘the integral part of ’). In fact, when x is itself an integer, these two bounds for
are identical.
As a numerical example, we seek an estimate for Ei(5)—and since our asymptotic
expansion is valid as x = 5 appears to be a rather bold choice. The remainder
then satisfies

and

i.e. 0.166 < I(5) < 0.174, where we have re-introduced the sign of the remainder, so
that and then we obtain 0.00112 < Ei(5) < 0.00117. The sur-
prise, perhaps, is that a divergent asymptotic expansion, valid as can produce
tolerable estimates for xs as small as 5. Of course, for larger values of x, the estimates
are more accurate e.g. 0.09155 < I(10) < 0.09158, from which we can obtain a good
estimate for Ei(10). Two further examples for you to investigate, similar to this one,
can be found in Q1. 11, 1.12; other asymptotic expansions of integrals are discussed
in Q1.13–1.17 and finding an expansion from a differential equation is the exercise in
Q1.18.

In this example, E1.5, we have used the alternating-sign property, but we could have
worked directly with the remainder, If it is possible to obtain a reasonable