7.1 Asymptotic Sequences and Expansions
Definition 7.1.1 (Order symbols). As \(x\to x_0\) we write \(f = O(g)\) if \(|f/g|\) is bounded, and \(f = o(g)\) if \(f/g\to 0\).
Definition 7.1.2 (Asymptotic sequence). A sequence \(\{\varphi _n\}\) is an asymptotic sequence as \(x\to x_0\) if \(\varphi _{n+1} = o(\varphi _n)\) for every \(n\). The usual choices are \(\varphi _n(x)=x^{n}\) as \(x\to 0\) and \(\varphi _n(x)=x^{-n}\) as \(x\to \infty \).
Definition 7.1.3 (Asymptotic expansion, Poincaré). \(f\) has the asymptotic expansion \(\sum a_n\varphi _n\), written \[f(x)\ \sim \ \sum ^{\infty }_{n=0}a_n\varphi _n(x),\] if for every \(N\) \[f(x) - \sum ^{N}_{n=0}a_n\varphi _n(x) = o\left (\varphi _N(x)\right ) \qquad \text {as } x\to x_0 .\]
Note 7.1.4. Read the definition carefully, because it is not the definition of a convergent series and the difference is the whole point. Convergence fixes \(x\) and lets \(N\to \infty \); an asymptotic expansion fixes \(N\) and lets \(x\to x_0\). A series may therefore be asymptotic and divergent at every \(x\), and the most useful ones usually are.
The practical consequence is that adding terms does not always improve matters. For a divergent asymptotic series there is an optimal number of terms, after which the partial sums get worse; the best accuracy attainable is roughly the size of the smallest term, and one stops there.
Example 7.1.5 (The exponential integral). Integrating by parts repeatedly, \[\int ^{\infty }_{x}\frac {e^{-t}}{t}\,dt = e^{-x}\left [\frac {1}{x} - \frac {1}{x^{2}} + \frac {2!}{x^{3}} - \frac {3!}{x^{4}} + \cdots + \frac {(-1)^{n}n!}{x^{n+1}}\right ] + R_n,\] with \(\left |R_n\right | \leq (n+1)!\,e^{-x}/x^{n+2}\). For fixed \(x\) the terms \(n!/x^{n+1}\) eventually grow without bound, so the series diverges for every \(x\). Yet at \(x=10\) the first four terms give the integral to five significant figures.
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