Oh I see. Thanks. But why does it not work when I try to evaluate the expansion directly?
You mean write the series expansion, for , where ? Then you have to justify that the last series converges to 0 when . There are ways to do that (this is an alternate series so we can get a uniform bound for the remainder of the series by Leibniz theorem), but it is not trivial, and first of all the asymptotic expansion is so much more appropriate (it is dedicated to computation of limits).